As Logical Reasoning remains a critical skill in problem-solving, decision-making, and analytical thinking, recruiters need to identify candidates who can think critically and solve complex challenges efficiently. Whether hiring for technical, analytical, or managerial roles, assessing logical reasoning ensures candidates possess the cognitive abilities required for success.
This resource, "100+ Logical Reasoning Questions and Answers," is designed to help recruiters evaluate candidates effectively. It covers various reasoning types, from fundamental to advanced concepts, including pattern recognition, syllogisms, puzzles, coding-decoding, and data interpretation.
Whether hiring for entry-level positions or leadership roles, this guide enables you to assess a candidate’s:
- Core Logical Reasoning Skills: Deductive and inductive reasoning, analogies, and number series.
- Advanced Problem-Solving Abilities: Critical thinking, statement-conclusion analysis, and logical puzzles.
- Real-World Application: Decision-making under constraints, problem-solving speed, and strategic thinking.
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Save time, improve candidate evaluation, and confidently hire professionals with strong analytical and logical reasoning abilities to drive business success.
Easy Level
- If all roses are flowers, and some flowers fade quickly, can we say that some roses fade quickly?
- A is taller than B, and B is taller than C. Who is the shortest?
- If it takes five workers three hours to complete a task, how long would it take ten workers to complete the same task?
- All cats are mammals. If some mammals are not dogs, can we conclude that some cats are not dogs?
- If a car travels 60 miles in 1 hour, how far will it travel in 4 hours at the same speed?
- John has two more apples than Mary. If Mary has 3 apples, how many apples does John have?
- If the day after tomorrow is Friday, what day is today?
- There are five apples in a basket. If you take away three, how many do you have?
- If a rectangle has a length of 8 and a width of 4, what is its perimeter?
- If all dogs bark and Max is a dog, what can we say about Max?
- A train leaves the station at 2 PM and travels at a speed of 50 mph. What time will it arrive if it travels for 2 hours?
- If the first day of the month is a Sunday, what day of the week is the 15th?
- You have a box with 10 chocolates. If you eat 2, how many are left?
- If it is raining, the ground is wet. It is raining. What can we conclude about the ground?
- In a class of 30 students, if 18 are girls, how many are boys?
- If a clock shows 3:15, what is the angle between the hour and minute hands?
- If it takes 2 minutes to boil an egg, how long will it take to boil 6 eggs?
- If you multiply a number by 2 and then subtract 3, you get 7. What is the number?
- A group of 4 friends can finish a project in 8 hours. How long will it take 8 friends to finish the same project?
- If you flip a coin twice, what is the probability of getting at least one head?
- A book has 200 pages. If you read 20 pages a day, how many days will it take to finish the book?
- If all humans are mammals, and some mammals are whales, can we conclude that some humans are whales?
- If the sum of two numbers is 20 and one of the numbers is 8, what is the other number?
- If you buy 3 apples for $1 each, how much do you spend?
- A farmer has 10 cows and 20 sheep. How many animals does he have in total?
- If a dozen eggs costs $2, how much would 3 dozen cost?
- If a train travels at 60 mph for 2 hours, how far does it travel?
- If today is Monday, what day will it be in three days?
- If you have 5 pairs of socks, how many individual socks do you have?
- If a rectangle has an area of 20 square units and a width of 5 units, what is its length?
- If 7 + x = 10, what is the value of x?
- If it takes 10 minutes to wash one car, how long will it take to wash 5 cars?
- If a pizza is cut into 8 slices and you eat 2 slices, how many slices are left?
- If a triangle has angles of 90 and 45 degrees, what is the third angle?
- If you have a jar with 5 red marbles and 3 blue marbles, what is the probability of picking a red marble?
- If a student scores 80 out of 100 in a test, what is the percentage score?
- If you multiply a number by 4 and the result is 28, what is the number?
- If it takes 30 minutes to travel 15 miles, what is the average speed in miles per hour?
- If the product of two numbers is 24 and one number is 4, what is the other number?
- If the weather is sunny, people go to the beach. Today is sunny. Where do people go?
Intermediate Level
- A woman is twice as old as her son. If the son is 10 years old, how old is the woman?
- If some A are B, and some B are C, can we conclude that some A are C?
- In a group of 100 people, 60 like tea and 40 like coffee. How many like both?
- If a cyclist travels 15 miles at a speed of 5 mph, how long does the journey take?
- A clock is 10 minutes fast. If it shows 2 PM, what is the actual time?
- If two trains leave the station at the same time and travel in opposite directions at 60 mph and 80 mph, how far apart will they be after 1 hour?
- If five times a number is 25, what is the number?
- In a race of 100 meters, if a runner completes the distance in 12 seconds, what is their speed in meters per second?
- If a school has 500 students, and 60% are girls, how many boys are there?
- A bag contains 5 red, 3 blue, and 2 green balls. What is the probability of picking a blue ball?
- If a car's fuel efficiency is 25 miles per gallon, how many gallons are needed to travel 100 miles?
- If x + 5 = 15, what is the value of x?
- If a man is 4 times as old as his son, and their combined age is 40, how old is the son?
- If it takes 3 hours to paint a house, how long will it take to paint 4 houses?
- A box contains 12 balls. If 3 balls are removed, what fraction of the balls remains in the box?
- If the ratio of cats to dogs is 3:2 and there are 12 cats, how many dogs are there?
- A store sells 10 apples for $3. How much will 25 apples cost?
- If a car travels 180 miles in 3 hours, what is its average speed?
- If the sum of three consecutive integers is 48, what are the integers?
- A recipe requires 2 cups of flour for every 3 cups of sugar. If you use 6 cups of flour, how much sugar do you need?
- If a rectangle has a length of 10 and a width of 6, what is its area?
- If the probability of an event is 0.2, what is the probability that the event does not occur?
- If a train travels at 70 mph, how long will it take to cover 280 miles?
- If the angles of a triangle are in the ratio 2:3:5, what is the measure of the largest angle?
- A person has $100 and spends $25. What percentage of the original amount is left?
- If it takes 4 workers 6 days to complete a project, how long will it take 6 workers to complete the same project?
- If a rectangular garden has a length of 12 meters and a width of 5 meters, what is the perimeter?
- If a man earns $500 a week, how much will he earn in a month (4 weeks)?
- A number is increased by 20% and then decreased by 20%. Is the final value greater than, less than, or equal to the original value?
- If a recipe needs 3 eggs to make 12 cookies, how many eggs are needed for 36 cookies?
- If a train leaves a station at 2 PM and arrives at 4 PM, how long is the journey?
- A bicycle costs $100, and there is a 20% discount. What is the final price?
- If a box contains 6 red balls and 4 green balls, what is the probability of picking a red ball?
- If the temperature is 20 degrees Celsius and rises by 15 degrees, what is the new temperature?
- If a pair of shoes costs $50 after a 25% discount, what was the original price?
- A class has 30 students, and 12 are boys. What percentage of the class are girls?
- If the product of two numbers is 36 and one number is 6, what is the other number?
- If a cake is divided into 8 equal slices and 3 slices are eaten, what fraction of the cake remains?
- If the radius of a circle is 7 cm, what is its circumference (use π = 3.14)?
- If you throw a die, what is the probability of rolling a number greater than 4?
Experience Level
- In a survey, 70% of people prefer tea over coffee, while 30% prefer coffee. If 200 people participated, how many prefer tea?
- If the sum of the angles in a triangle is 180 degrees, what is the measure of the third angle if the other two are 60 and 70 degrees?
- A man has twice as many apples as his friend. If together they have 36 apples, how many apples does each have?
- If the cost of 3 notebooks is $9, what is the cost of 8 notebooks?
- If a sum of money is divided among A, B, and C in the ratio 3:4:5, what fraction does B receive if the total amount is $120?
- If a rectangular prism has a length of 10 cm, a width of 5 cm, and a height of 2 cm, what is its volume?
- If an investment of $1,000 earns 5% interest per year, how much will it be worth after 3 years?
- In a class of 50 students, 30 passed Math, 25 passed Science, and 15 passed both. How many students passed at least one subject?
- A factory produces 500 toys in 8 hours. How many toys can it produce in 12 hours at the same rate?
- If a box contains 20 balls: 10 red, 5 blue, and 5 green, what is the probability of randomly selecting a blue ball?
- If a car travels 150 miles on 5 gallons of gas, what is its mileage in miles per gallon?
- In a family of six members, the average age is 30 years. What is the total age of all family members?
- A bag contains 12 white balls and 8 black balls. What is the probability of picking a white ball?
- If the ratio of boys to girls in a class is 4:5 and there are 36 students in total, how many girls are there?
- If a shopkeeper sells an item for $120 after a 25% discount, what was the original price?
- If the sides of a square are doubled, what happens to the area?
- If a person can complete a task in 10 hours, how much of the task can they complete in 4 hours?
- If the median of the numbers 3, 7, 9, and x is 7, what can we say about the value of x?
- A computer can process 1000 operations in a second. How many operations can it process in 5 minutes?
- If a triangle has sides of lengths 5, 12, and 13, is it a right triangle?
- If the probability of winning a game is 0.4, what is the probability of losing?
- If the ages of two siblings are in the ratio 3:4 and the sum of their ages is 28, how old is the older sibling?
- A train travels 300 miles at a speed of 75 mph. How long does it take to complete the journey?
- If a triangle has two angles measuring 50 and 60 degrees, what is the measure of the third angle?
- In a class, 60% of students passed Mathematics, and 50% passed Science. If 30% passed both, what percentage passed at least one subject?
- If the area of a circle is 78.5 square cm, what is its radius (use π = 3.14)?
- If a rectangular garden is 15 meters long and 10 meters wide, what is its area?
- If the product of three consecutive integers is 210, what are the integers?
- If a stock price increases from $50 to $75, what is the percentage increase?
- In a box with 3 red, 4 green, and 5 blue marbles, what is the probability of picking a green marble?
- If you flip a coin three times, what is the probability of getting exactly two heads?
- If a person earns $200 per day, how much will they earn in a 30-day month?
- If a bag contains 4 apples, 6 bananas, and 10 oranges, what is the probability of randomly picking an apple?
- If the ages of three friends are in the ratio 2:3:4 and the total age is 72, what is the age of the oldest friend?
- If the sum of three consecutive even numbers is 54, what are the numbers?
- If a train leaves a station at 10 AM and travels at 60 mph, what time will it arrive at a destination 180 miles away?
- If a company produces 200 units of a product and sells each for $50, what is the total revenue?
- If the circumference of a circle is 31.4 cm, what is its diameter (use π = 3.14)?
- If a student scored 85 out of 100 in one test and 90 out of 100 in another, what is their average score?
- If an item costs $80 after a 20% increase, what was its original price?
Ans. Easy level
1. If all roses are flowers, and some flowers fade quickly, can we say that some roses fade quickly?
No, we cannot conclude that some roses fade quickly. The premise states that "all roses are flowers," which places roses within the larger category of flowers. However, the statement that "some flowers fade quickly" does not specify which types of flowers fade quickly. There may be roses among those flowers, or there may not be. Thus, while some flowers may fade quickly, it does not imply that roses are included in that group.
2. A is taller than B, and B is taller than C. Who is the shortest?
C is the shortest.
- According to the information given:some text
- A > B (A is taller than B)
- B > C (B is taller than C)
From these two relationships, we can deduce the following order of height:
A > B > C.
Therefore, C is the shortest of the three individuals.
3. If it takes five workers three hours to complete a task, how long would it take ten workers to complete the same task?
It would take six workers 1.5 hours to complete the task.
- To find out how long it will take ten workers, we first calculate the total work done in worker-hours:some text
- Total work = Number of workers × Time taken = 5 workers × 3 hours = 15 worker-hours.
Now, if we have ten workers, we can find the time taken to complete the same work:
- Time = Total work / Number of workers = 15 worker-hours / 10 workers = 1.5 hours.
Thus, ten workers can complete the task in 1.5 hours.
4. All cats are mammals. If some mammals are not dogs, can we conclude that some cats are not dogs?
Yes, we can conclude that some cats are not dogs.
- The premises state:some text
- All cats are mammals (which means every cat belongs to the category of mammals).
- Some mammals are not dogs.
Since some mammals are not dogs, and since all cats fall under the category of mammals, it is logically possible that those cats that are mammals may be among those that are not dogs. Therefore, we can conclude that some cats are not dogs.
5. If a car travels 60 miles in 1 hour, how far will it travel in 4 hours at the same speed?
It will travel 240 miles.
- The speed of the car is given as 60 miles per hour. To find the distance traveled in 4 hours, we can use the formula:
Distance = Speed × Timesome text- Distance = 60 miles/hour × 4 hours = 240 miles.
So, the car will travel 240 miles in 4 hours.
6. John has two more apples than Mary. If Mary has 3 apples, how many apples does John have?
John has 5 apples.
- We know that:some text
- Mary has 3 apples.
- John has two more apples than Mary, which can be expressed as:
John's apples = Mary's apples + 2 - Substituting the number of apples Mary has:
John's apples = 3 + 2 = 5
Thus, John has 5 apples.
7. If the day after tomorrow is Friday, what day is today?
Today is Wednesday.
- If the day after tomorrow is Friday, then:some text
- Tomorrow will be Thursday.
- Therefore, today is Wednesday.
So, today is Wednesday.
8. There are five apples in a basket. If you take away three, how many do you have?
You have three apples.
- The question states that if you take away three apples from the basket, you possess those three apples. It does not ask how many apples are left in the basket; it asks how many you have taken.
- Therefore, the answer is that you have 3 apples.
9. If a rectangle has a length of 8 and a width of 4, what is its perimeter?
The perimeter is 24.
- The formula for calculating the perimeter (P) of a rectangle is:
P = 2 × (Length + Width) - Substituting the values:some text
- Length = 8
- Width = 4
- Perimeter = 2 × (8 + 4) = 2 × 12 = 24.
Thus, the perimeter of the rectangle is 24.
10. If all dogs bark and Max is a dog, what can we say about Max?
Max barks.
- The premises are:some text
- All dogs bark (this establishes a general rule).
- Max is a dog (this identifies Max as part of that category).
- Since Max is included in the group of all dogs, we can conclude that he exhibits the characteristic of barking. Therefore, we can say that Max barks.
11. A train leaves the station at 2 PM and travels at a speed of 50 mph. What time will it arrive if it travels for 2 hours?
The train will arrive at 4 PM.
- Departure time: 2 PM
- Travel time: 2 hours
- Arrival time: 2 PM+2 hours=4 PM2 \text{ PM} + 2 \text{ hours} = 4 \text{ PM}2 PM+2 hours=4 PM
12. If the first day of the month is a Sunday, what day of the week is the 15th?
The 15th day of the month will be a Sunday.
- Counting from the 1st (Sunday):some text
- 1st: Sunday
- 2nd: Monday
- 3rd: Tuesday
- 4th: Wednesday
- 5th: Thursday
- 6th: Friday
- 7th: Saturday
- 8th: Sunday
- 9th: Monday
- 10th: Tuesday
- 11th: Wednesday
- 12th: Thursday
- 13th: Friday
- 14th: Saturday
- 15th: Sunday
So, the 15th is also a Sunday.
13. You have a box with 10 chocolates. If you eat 2, how many are left?
You have 8 chocolates left.
- Initial chocolates: 10
- Chocolates eaten: 2
- Remaining chocolates: 10−2=810 - 2 = 810−2=8
Thus, there are 8 chocolates left.
14. If it is raining, the ground is wet. It is raining. What can we conclude about the ground?
We can conclude that the ground is wet.
- This follows from the logical premise:some text
- If it is raining, then the ground is wet.
- Since it is raining, we conclude that the ground must be wet.
15. In a class of 30 students, if 18 are girls, how many are boys?
There are 12 boys in the class.
- Total students: 30
- Girls: 18
- Boys: 30−18=1230 - 18 = 1230−18=12
Thus, there are 12 boys.
16. If a clock shows 3:15, what is the angle between the hour and minute hands?
The angle between the hour and minute hands is 52.5 degrees.
- The minute hand at 15 minutes is at the 3 on the clock. Each minute represents 666 degrees (since 360/60=6360/60 = 6360/60=6). So the minute hand is at:
15×6=90 degrees15 \times 6 = 90 \text{ degrees}15×6=90 degrees - The hour hand moves 303030 degrees per hour (since 360/12=30360/12 = 30360/12=30). At 3:15, the hour hand has moved:
3×30+(1560×30)=90+7.5=97.5 degrees3 \times 30 + \left( \frac{15}{60} \times 30 \right) = 90 + 7.5 = 97.5 \text{ degrees}3×30+(6015×30)=90+7.5=97.5 degrees - Therefore, the angle between the hour and minute hands is:
∣97.5−90∣=7.5 degrees|97.5 - 90| = 7.5 \text{ degrees}∣97.5−90∣=7.5 degrees
So the angle is 7.5 degrees.
17. If it takes 2 minutes to boil an egg, how long will it take to boil 6 eggs?
It takes 2 minutes to boil all 6 eggs.
- Assuming you can boil all 6 eggs at once, it will still only take 2 minutes.
So, it will take 2 minutes.
18. If you multiply a number by 2 and then subtract 3, you get 7. What is the number?
The number is 5.
- Let the number be xxx.
- The equation is:
2x−3=72x - 3 = 72x−3=7 - Adding 3 to both sides:
2x=102x = 102x=10 - Dividing by 2:
x=5x = 5x=5
Thus, the number is 5.
19. A group of 4 friends can finish a project in 8 hours. How long will it take 8 friends to finish the same project?
It will take 4 hours for 8 friends to finish the project.
- If 4 friends can complete the project in 8 hours, their combined work rate is 18\frac{1}{8}81 of the project per hour.
- Doubling the number of friends to 8 means they will complete the project at twice the rate:
8 friends=2×18=14 of the project per hour8 \text{ friends} = 2 \times \frac{1}{8} = \frac{1}{4} \text{ of the project per hour}8 friends=2×81=41 of the project per hour - Therefore, it will take 444 hours to finish the project:
1 project14 project/hour=4 hours\frac{1 \text{ project}}{\frac{1}{4} \text{ project/hour}} = 4 \text{ hours}41 project/hour1 project=4 hours
So, it will take 4 hours.
20. If you flip a coin twice, what is the probability of getting at least one head?
The probability of getting at least one head is 34\frac{3}{4}43 or 75%.
- The possible outcomes when flipping a coin twice are:some text
- HH (two heads)
- HT (one head, one tail)
- TH (one tail, one head)
- TT (two tails)
- Out of these 4 outcomes, 3 of them (HH, HT, TH) contain at least one head.
- Therefore, the probability is:
P(at least one head)=34P(\text{at least one head}) = \frac{3}{4}P(at least one head)=43
Thus, the probability is 34\frac{3}{4}43.
21. A book has 200 pages. If you read 20 pages a day, how many days will it take to finish the book?
It will take 10 days to finish the book.
- Total pages: 200
- Pages read per day: 20
- Days to finish the book:
200 pages20 pages/day=10 days\frac{200 \text{ pages}}{20 \text{ pages/day}} = 10 \text{ days}20 pages/day200 pages=10 days
Thus, it will take 10 days.
22. If all humans are mammals, and some mammals are whales, can we conclude that some humans are whales?
No, we cannot conclude that some humans are whales.
- The logical reasoning is as follows:some text
- All humans are indeed mammals.
- Some mammals are whales, but this does not imply that humans are among those mammals.
- Therefore, we cannot conclude that some humans are whales.
23. If the sum of two numbers is 20 and one of the numbers is 8, what is the other number?
The other number is 12.
- Let the two numbers be xxx and yyy.
- Given:
x+y=20x + y = 20x+y=20
x=8x = 8x=8 - To find yyy:
8+y=208 + y = 208+y=20
y=20−8=12y = 20 - 8 = 12y=20−8=12
Thus, the other number is 12.
24. If you buy 3 apples for $1 each, how much do you spend?
You spend $3.
- Cost per apple: $1
- Number of apples: 3
- Total cost:
3 apples×1 dollar/apple=3 dollars3 \text{ apples} \times 1 \text{ dollar/apple} = 3 \text{ dollars}3 apples×1 dollar/apple=3 dollars
So, you spend $3.
25. A farmer has 10 cows and 20 sheep. How many animals does he have in total?
The farmer has 30 animals in total.
- Cows: 10
- Sheep: 20
- Total animals:
10+20=3010 + 20 = 3010+20=30
Thus, the farmer has 30 animals.
26. If a dozen eggs costs $2, how much would 3 dozen cost?
3 dozen eggs would cost $6.
- Cost of 1 dozen: $2
- Cost of 3 dozen:
3×2=6 dollars3 \times 2 = 6 \text{ dollars}3×2=6 dollars
So, it would cost $6.
27. If a train travels at 60 mph for 2 hours, how far does it travel?
The train travels 120 miles.
- Speed: 60 mph
- Time: 2 hours
- Distance traveled:
Distance=Speed×Time=60 mph×2 hours=120 miles\text{Distance} = \text{Speed} \times \text{Time} = 60 \text{ mph} \times 2 \text{ hours} = 120 \text{ miles}Distance=Speed×Time=60 mph×2 hours=120 miles
Thus, it travels 120 miles.
28. If today is Monday, what day will it be in three days?
It will be Thursday in three days.
- Starting from Monday:some text
- Day 1: Tuesday
- Day 2: Wednesday
- Day 3: Thursday
So, it will be Thursday.
29. If you have 5 pairs of socks, how many individual socks do you have?
You have 10 individual socks.
- Pairs of socks: 5
- Individual socks:
5 pairs×2=10 individual socks5 \text{ pairs} \times 2 = 10 \text{ individual socks}5 pairs×2=10 individual socks
Thus, you have 10 individual socks.
30. If a rectangle has an area of 20 square units and a width of 5 units, what is its length?
The length of the rectangle is 4 units.
- Area formula:
Area=Length×Width\text{Area} = \text{Length} \times \text{Width}Area=Length×Width - Given area: 20 square units and width: 5 units:
20=Length×520 = \text{Length} \times 520=Length×5 - Solving for length:
Length=205=4 units\text{Length} = \frac{20}{5} = 4 \text{ units}Length=520=4 units
Thus, the length is 4 units.
31. If 7+x=107 + x = 107+x=10, what is the value of xxx?
The value of xxx is 3.
- Starting with the equation:
7+x=107 + x = 107+x=10 - To find xxx, subtract 7 from both sides:
x=10−7=3x = 10 - 7 = 3x=10−7=3
Thus, xxx is 3.
32. If it takes 10 minutes to wash one car, how long will it take to wash 5 cars?
It will take 50 minutes to wash 5 cars.
- Time to wash one car: 10 minutes
- Total time for 5 cars:
5 cars×10 minutes/car=50 minutes5 \text{ cars} \times 10 \text{ minutes/car} = 50 \text{ minutes}5 cars×10 minutes/car=50 minutes
So, it will take 50 minutes.
33. If a pizza is cut into 8 slices and you eat 2 slices, how many slices are left?
There are 6 slices left.
- Total slices: 8
- Slices eaten: 2
- Remaining slices:
8−2=68 - 2 = 68−2=6
Thus, there are 6 slices left.
34. If a triangle has angles of 90 and 45 degrees, what is the third angle?
The third angle is 45 degrees.
- The sum of angles in a triangle is 180 degrees:
90+45+third angle=18090 + 45 + \text{third angle} = 18090+45+third angle=180 - Solving for the third angle:
third angle=180−90−45=45 degrees\text{third angle} = 180 - 90 - 45 = 45 \text{ degrees}third angle=180−90−45=45 degrees
Thus, the third angle is 45 degrees.
35. If you have a jar with 5 red marbles and 3 blue marbles, what is the probability of picking a red marble?
The probability of picking a red marble is 58\frac{5}{8}85.
- Total marbles: 5+3=85 + 3 = 85+3=8
- Red marbles: 5
- Probability:
P(red)=Number of red marblesTotal number of marbles=58P(\text{red}) = \frac{\text{Number of red marbles}}{\text{Total number of marbles}} = \frac{5}{8}P(red)=Total number of marblesNumber of red marbles=85
Thus, the probability of picking a red marble is 58\frac{5}{8}85.
36. If a student scores 80 out of 100 in a test, what is the percentage score?
The percentage score is 80%.
- Percentage formula:
Percentage=(ScoreTotal Score)×100\text{Percentage} = \left( \frac{\text{Score}}{\text{Total Score}} \right) \times 100Percentage=(Total ScoreScore)×100 - Calculation:
Percentage=(80100)×100=80%\text{Percentage} = \left( \frac{80}{100} \right) \times 100 = 80\%Percentage=(10080)×100=80%
Thus, the percentage score is 80%.
37. If you multiply a number by 4 and the result is 28, what is the number?
The number is 7.
- Let the number be xxx.
- Given:
4x=284x = 284x=28 - To find xxx, divide both sides by 4:
x=284=7x = \frac{28}{4} = 7x=428=7
Thus, the number is 7.
38. If it takes 30 minutes to travel 15 miles, what is the average speed in miles per hour?
The average speed is 30 mph.
- Speed formula:
Speed=DistanceTime\text{Speed} = \frac{\text{Distance}}{\text{Time}}Speed=TimeDistance - Distance: 15 miles
- Time: 30 minutes (which is 3060=0.5\frac{30}{60} = 0.56030=0.5 hours)
- Average speed:
Speed=15 miles0.5 hours=30 mph\text{Speed} = \frac{15 \text{ miles}}{0.5 \text{ hours}} = 30 \text{ mph}Speed=0.5 hours15 miles=30 mph
Thus, the average speed is 30 mph.
39. If the product of two numbers is 24 and one number is 4, what is the other number?
The other number is 6.
- Let the two numbers be xxx and yyy.
- Given:
xy=24xy = 24xy=24
x=4x = 4x=4 - To find yyy:
4y=244y = 244y=24
y=244=6y = \frac{24}{4} = 6y=424=6
Thus, the other number is 6.
40. If the weather is sunny, people go to the beach. Today is sunny. Where do people go?
People go to the beach.
- The statement indicates a conditional relationship: if it is sunny, then people go to the beach.
- Since today is sunny, we conclude that people will go to the beach.
Thus, people go to the beach.
Intermediate level ans.
1. A woman is twice as old as her son. If the son is 10 years old, how old is the woman?
The woman is 20 years old.
- If the son is 10 years old, then:
Woman’s age=2×Son’s age=2×10=20\text{Woman's age} = 2 \times \text{Son's age} = 2 \times 10 = 20Woman’s age=2×Son’s age=2×10=20
So, the woman is 20 years old.
2. If some A are B, and some B are C, can we conclude that some A are C?
No, we cannot conclude that some A are C.
- Just because some A are B and some B are C does not imply that there is an overlap between A and C.
- The relationship does not guarantee that any of the A that are B also intersect with the C group.
3. In a group of 100 people, 60 like tea and 40 like coffee. How many like both?
To find out how many like both, we can use the principle of inclusion-exclusion.
Let xxx be the number of people who like both tea and coffee.
- Total who like tea: 60
- Total who like coffee: 40
- Total number of people: 100
Using the formula:
Total=(Tea)+(Coffee)−(Both)\text{Total} = (\text{Tea}) + (\text{Coffee}) - (\text{Both}) Total=(Tea)+(Coffee)−(Both) 100=60+40−x100 = 60 + 40 - x 100=60+40−x 100=100−x⇒x=0100 = 100 - x \Rightarrow x = 0100=100−x⇒x=0
Thus, 0 people like both tea and coffee.
4. If a cyclist travels 15 miles at a speed of 5 mph, how long does the journey take?
The journey takes 3 hours.
- Time is calculated by the formula:
Time=DistanceSpeed\text{Time} = \frac{\text{Distance}}{\text{Speed}} Time=SpeedDistance
Time=15 miles5 mph=3 hours\text{Time} = \frac{15 \text{ miles}}{5 \text{ mph}} = 3 \text{ hours} Time=5 mph15 miles=3 hours
Thus, the journey takes 3 hours.
5. A clock is 10 minutes fast. If it shows 2 PM, what is the actual time?
The actual time is 1:50 PM.
- If the clock is 10 minutes fast, we subtract 10 minutes from the displayed time.
- Thus,
2:00 PM−10 minutes=1:50 PM2:00 \text{ PM} - 10 \text{ minutes} = 1:50 \text{ PM} 2:00 PM−10 minutes=1:50 PM
So, the actual time is 1:50 PM.
6. If two trains leave the station at the same time and travel in opposite directions at 60 mph and 80 mph, how far apart will they be after 1 hour?
They will be 140 miles apart.
- The distance each train travels in one hour:some text
- Train 1: 60 miles
- Train 2: 80 miles
- Total distance apart:
60 miles+80 miles=140 miles60 \text{ miles} + 80 \text{ miles} = 140 \text{ miles} 60 miles+80 miles=140 miles
Thus, they will be 140 miles apart.
7. If five times a number is 25, what is the number?
The number is 5.
5x=255x = 25 5x=25
x=255=5x = \frac{25}{5} = 5 x=525=5
So, the number is 5.
8. In a race of 100 meters, if a runner completes the distance in 12 seconds, what is their speed in meters per second?
The speed is 8.33 meters per second.
Speed=DistanceTime\text{Speed} = \frac{\text{Distance}}{\text{Time}} Speed=TimeDistance
Speed=100 meters12 seconds≈8.33 m/s\text{Speed} = \frac{100 \text{ meters}}{12 \text{ seconds}} \approx 8.33 \text{ m/s} Speed=12 seconds100 meters≈8.33 m/s
Thus, the speed is 8.33 m/s.
9. If a school has 500 students, and 60% are girls, how many boys are there?
There are 200 boys.
- Total students: 500
- Percentage of girls: 60%
- Number of girls:
Girls=0.6×500=300\text{Girls} = 0.6 \times 500 = 300 Girls=0.6×500=300
Boys=500−300=200\text{Boys} = 500 - 300 = 200 Boys=500−300=200
Thus, there are 200 boys.
10. A bag contains 5 red, 3 blue, and 2 green balls. What is the probability of picking a blue ball?
The probability of picking a blue ball is 310\frac{3}{10}103.
5+3+2=105 + 3 + 2 = 10 5+3+2=10
Probability of blue=Number of blue ballsTotal number of balls=310\text{Probability of blue} = \frac{\text{Number of blue balls}}{\text{Total number of balls}} = \frac{3}{10} Probability of blue=Total number of ballsNumber of blue balls=103
Thus, the probability of picking a blue ball is 310\frac{3}{10}103.
11. If a car's fuel efficiency is 25 miles per gallon, how many gallons are needed to travel 100 miles?
You will need 4 gallons.
- Fuel efficiency: 25 miles/gallon
- Distance to travel: 100 miles
- Calculation:
Gallons needed=DistanceFuel efficiency=100 miles25 miles/gallon=4 gallons\text{Gallons needed} = \frac{\text{Distance}}{\text{Fuel efficiency}} = \frac{100 \text{ miles}}{25 \text{ miles/gallon}} = 4 \text{ gallons}Gallons needed=Fuel efficiencyDistance=25 miles/gallon100 miles=4 gallons
Thus, you will need 4 gallons.
12. If x+5=15x + 5 = 15x+5=15, what is the value of xxx?
The value of xxx is 10.
x+5=15x + 5 = 15 x+5=15
Subtracting 5 from both sides:
x=15−5=10x = 15 - 5 = 10 x=15−5=10
So, x=10x = 10x=10.
13. If a man is 4 times as old as his son, and their combined age is 40, how old is the son?
The son is 8 years old.
- Let the son's age be xxx. Then the man's age is 4x4x4x.
- Combined age:
x+4x=40x + 4x = 40 x+4x=40
5x=40⇒x=85x = 40 \Rightarrow x = 8 5x=40⇒x=8
So, the son is 8 years old.
14. If it takes 3 hours to paint a house, how long will it take to paint 4 houses?
It will take 12 hours to paint 4 houses.
- Time to paint one house: 3 hours
- Total time for 4 houses:
4 houses×3 hours/house=12 hours4 \text{ houses} \times 3 \text{ hours/house} = 12 \text{ hours} 4 houses×3 hours/house=12 hours
Thus, it will take 12 hours.
15. A box contains 12 balls. If 3 balls are removed, what fraction of the balls remains in the box?
The fraction of balls remaining is 912\frac{9}{12}129 or 34\frac{3}{4}43.
- Initial number of balls: 12
- Balls removed: 3
- Remaining balls:
12−3=912 - 3 = 9 12−3=9
912=34\frac{9}{12} = \frac{3}{4} 129=43
Thus, the fraction of balls remaining is 34\frac{3}{4}43.
16. If the ratio of cats to dogs is 3:2 and there are 12 cats, how many dogs are there?
There are 8 dogs.
- Given ratio of cats to dogs is 3:2.
- Let the number of dogs be xxx:
32=12x\frac{3}{2} = \frac{12}{x} 23=x12
Cross-multiplying:
3x=24⇒x=83x = 24 \Rightarrow x = 8 3x=24⇒x=8
Thus, there are 8 dogs.
17. A store sells 10 apples for $3. How much will 25 apples cost?
The cost for 25 apples will be $7.50.
310=0.30 dollars per apple\frac{3}{10} = 0.30 \text{ dollars per apple} 103=0.30 dollars per apple
0.30×25=7.500.30 \times 25 = 7.50 0.30×25=7.50
Thus, the cost for 25 apples is $7.50.
18. If a car travels 180 miles in 3 hours, what is its average speed?
The average speed is 60 mph.
Speed=DistanceTime\text{Speed} = \frac{\text{Distance}}{\text{Time}} Speed=TimeDistance
Speed=180 miles3 hours=60 mph\text{Speed} = \frac{180 \text{ miles}}{3 \text{ hours}} = 60 \text{ mph} Speed=3 hours180 miles=60 mph
Thus, the average speed is 60 mph.
19. If the sum of three consecutive integers is 48, what are the integers?
The integers are 15, 16, and 17.
- Let the first integer be xxx.
- The integers are x,x+1,x+2x, x + 1, x + 2x,x+1,x+2.
- Equation:
x+(x+1)+(x+2)=48x + (x + 1) + (x + 2) = 48 x+(x+1)+(x+2)=48 3x+3=48⇒3x=45⇒x=153x + 3 = 48 \Rightarrow 3x = 45 \Rightarrow x = 15 3x+3=48⇒3x=45⇒x=15
Thus, the integers are 15, 16, and 17.
20. A recipe requires 2 cups of flour for every 3 cups of sugar. If you use 6 cups of flour, how much sugar do you need?
You will need 9 cups of sugar.
- The ratio of flour to sugar is 23\frac{2}{3}32.
- If using 6 cups of flour:
Sugar needed=6 cups flour×3 cups sugar2 cups flour=9 cups sugar\text{Sugar needed} = 6 \text{ cups flour} \times \frac{3 \text{ cups sugar}}{2 \text{ cups flour}} = 9 \text{ cups sugar} Sugar needed=6 cups flour×2 cups flour3 cups sugar=9 cups sugar
Thus, you need 9 cups of sugar.
21. If a rectangle has a length of 10 and a width of 6, what is its area?
The area is 60 square units.
Area=Length×Width\text{Area} = \text{Length} \times \text{Width} Area=Length×Width
Area=10 units×6 units=60 square units\text{Area} = 10 \text{ units} \times 6 \text{ units} = 60 \text{ square units} Area=10 units×6 units=60 square units
Thus, the area of the rectangle is 60 square units.
22. If the probability of an event is 0.2, what is the probability that the event does not occur?
The probability that the event does not occur is 0.8.
- Probability of the event occurring: P(A)=0.2P(A) = 0.2P(A)=0.2
- Probability of the event not occurring:
P(not A)=1−P(A)=1−0.2=0.8P(\text{not A}) = 1 - P(A) = 1 - 0.2 = 0.8 P(not A)=1−P(A)=1−0.2=0.8
Thus, the probability that the event does not occur is 0.8.
23. If a train travels at 70 mph, how long will it take to cover 280 miles?
It will take 4 hours.
Time=DistanceSpeed\text{Time} = \frac{\text{Distance}}{\text{Speed}} Time=SpeedDistance
Time=280 miles70 mph=4 hours\text{Time} = \frac{280 \text{ miles}}{70 \text{ mph}} = 4 \text{ hours} Time=70 mph280 miles=4 hours
Thus, it will take 4 hours.
24. If the angles of a triangle are in the ratio 2:3:5, what is the measure of the largest angle?
The largest angle is 100 degrees.
- Let the angles be 2x,3x,2x, 3x,2x,3x, and 5x5x5x.
- Since the sum of angles in a triangle is 180 degrees:
2x+3x+5x=1802x + 3x + 5x = 180 2x+3x+5x=180 10x=180⇒x=1810x = 180 \Rightarrow x = 18 10x=180⇒x=18
5x=5×18=90 degrees5x = 5 \times 18 = 90 \text{ degrees} 5x=5×18=90 degrees
Thus, the largest angle is 90 degrees.
25. A person has $100 and spends $25. What percentage of the original amount is left?
The percentage left is 75%.
100−25=75100 - 25 = 75 100−25=75
- Percentage of original amount left:
75100×100=75%\frac{75}{100} \times 100 = 75\% 10075×100=75%
Thus, 75% of the original amount is left.
26. If it takes 4 workers 6 days to complete a project, how long will it take 6 workers to complete the same project?
It will take 4.8 days for 6 workers.
- Total work in worker-days:
4 workers×6 days=24 worker-days4 \text{ workers} \times 6 \text{ days} = 24 \text{ worker-days} 4 workers×6 days=24 worker-days
Time=24 worker-days6 workers=4 days\text{Time} = \frac{24 \text{ worker-days}}{6 \text{ workers}} = 4 \text{ days} Time=6 workers24 worker-days=4 days
Thus, it will take 4 days for 6 workers.
27. If a rectangular garden has a length of 12 meters and a width of 5 meters, what is the perimeter?
The perimeter is 34 meters.
Perimeter=2×(Length+Width)\text{Perimeter} = 2 \times (\text{Length} + \text{Width}) Perimeter=2×(Length+Width)
Perimeter=2×(12 m+5 m)=2×17 m=34 m\text{Perimeter} = 2 \times (12 \text{ m} + 5 \text{ m}) = 2 \times 17 \text{ m} = 34 \text{ m} Perimeter=2×(12 m+5 m)=2×17 m=34 m
Thus, the perimeter is 34 meters.
28. If a man earns $500 a week, how much will he earn in a month (4 weeks)?
He will earn $2000 in a month.
- Weekly earnings: $500
- Monthly earnings:
Earnings in a month=500 dollars/week×4 weeks=2000 dollars\text{Earnings in a month} = 500 \text{ dollars/week} \times 4 \text{ weeks} = 2000 \text{ dollars} Earnings in a month=500 dollars/week×4 weeks=2000 dollars
Thus, he will earn $2000 in a month.
29. A number is increased by 20% and then decreased by 20%. Is the final value greater than, less than, or equal to the original value?
The final value is less than the original value.
- Let the original number be xxx.
- After a 20% increase:
x+0.2x=1.2xx + 0.2x = 1.2x x+0.2x=1.2x
1.2x−0.2(1.2x)=1.2x−0.24x=0.96x1.2x - 0.2(1.2x) = 1.2x - 0.24x = 0.96x 1.2x−0.2(1.2x)=1.2x−0.24x=0.96x
Thus, the final value is less than the original value.
30. If a recipe needs 3 eggs to make 12 cookies, how many eggs are needed for 36 cookies?
You will need 9 eggs.
- Ratio of eggs to cookies:
3 eggs12 cookies=x eggs36 cookies\frac{3 \text{ eggs}}{12 \text{ cookies}} = \frac{x \text{ eggs}}{36 \text{ cookies}} 12 cookies3 eggs=36 cookiesx eggs
3×36=12x⇒108=12x⇒x=93 \times 36 = 12x \Rightarrow 108 = 12x \Rightarrow x = 9 3×36=12x⇒108=12x⇒x=9
Thus, you need 9 eggs for 36 cookies.
31. If a train leaves a station at 2 PM and arrives at 4 PM, how long is the journey?
The journey is 2 hours long.
Arrival time−Departure time=4 PM−2 PM=2 hours\text{Arrival time} - \text{Departure time} = 4 \text{ PM} - 2 \text{ PM} = 2 \text{ hours} Arrival time−Departure time=4 PM−2 PM=2 hours
Thus, the journey is 2 hours long.
32. A bicycle costs $100, and there is a 20% discount. What is the final price?
The final price is $80.
Discount=20% of 100=0.20×100=20 dollars\text{Discount} = 20\% \text{ of } 100 = 0.20 \times 100 = 20 \text{ dollars} Discount=20% of 100=0.20×100=20 dollars
Final price=100−20=80 dollars\text{Final price} = 100 - 20 = 80 \text{ dollars} Final price=100−20=80 dollars
Thus, the final price is $80.
33. If a box contains 6 red balls and 4 green balls, what is the probability of picking a red ball?
The probability of picking a red ball is 0.6.
6+4=106 + 4 = 10 6+4=10
P(Red)=Number of Red BallsTotal Number of Balls=610=0.6P(\text{Red}) = \frac{\text{Number of Red Balls}}{\text{Total Number of Balls}} = \frac{6}{10} = 0.6 P(Red)=Total Number of BallsNumber of Red Balls=106=0.6
Thus, the probability of picking a red ball is 0.6.
34. If the temperature is 20 degrees Celsius and rises by 15 degrees, what is the new temperature?
The new temperature is 35 degrees Celsius.
New Temperature=20+15=35 degrees Celsius\text{New Temperature} = 20 + 15 = 35 \text{ degrees Celsius} New Temperature=20+15=35 degrees Celsius
Thus, the new temperature is 35 degrees Celsius.
35. If a pair of shoes costs $50 after a 25% discount, what was the original price?
The original price is $66.67.
- Let the original price be xxx.
- After a 25% discount, the price is:
x−0.25x=0.75x=50x - 0.25x = 0.75x = 50 x−0.25x=0.75x=50
x=500.75=66.67x = \frac{50}{0.75} = 66.67 x=0.7550=66.67
Thus, the original price was $66.67.
36. A class has 30 students, and 12 are boys. What percentage of the class are girls?
60% of the class are girls.
30−12=1830 - 12 = 18 30−12=18
1830×100=60%\frac{18}{30} \times 100 = 60\% 3018×100=60%
Thus, 60% of the class are girls.
37. If the product of two numbers is 36 and one number is 6, what is the other number?
The other number is 6.
Other Number=366=6\text{Other Number} = \frac{36}{6} = 6 Other Number=636=6
Thus, the other number is 6.
38. If a cake is divided into 8 equal slices and 3 slices are eaten, what fraction of the cake remains?
The fraction of the cake that remains is 58\frac{5}{8}85.
8−3=58 - 3 = 5 8−3=5
58\frac{5}{8} 85
Thus, the fraction of the cake that remains is 58\frac{5}{8}85.
39. If the radius of a circle is 7 cm, what is its circumference (use π=3.14\pi = 3.14π=3.14)?
The circumference is 43.96 cm.
Circumference=2πr\text{Circumference} = 2\pi r Circumference=2πr
Circumference=2×3.14×7=43.96 cm\text{Circumference} = 2 \times 3.14 \times 7 = 43.96 \text{ cm} Circumference=2×3.14×7=43.96 cm
Thus, the circumference is 43.96 cm.
40. If you throw a die, what is the probability of rolling a number greater than 4?
The probability is 13\frac{1}{3}31.
- Favorable outcomes (rolling a 5 or 6): 2
- Total outcomes: 6
- Probability formula:
P(Rolling > 4)=Favorable OutcomesTotal Outcomes=26=13P(\text{Rolling > 4}) = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{2}{6} = \frac{1}{3} P(Rolling > 4)=Total OutcomesFavorable Outcomes=62=31
Thus, the probability of rolling a number greater than 4 is 13\frac{1}{3}31.
Experience level ans.
1. In a survey, 70% of people prefer tea over coffee, while 30% prefer coffee. If 200 people participated, how many prefer tea?
To find out how many people prefer tea:
Number of people who prefer tea=70% of 200=70100×200=140\text{Number of people who prefer tea} = 70\% \text{ of } 200 = \frac{70}{100} \times 200 = 140Number of people who prefer tea=70% of 200=10070×200=140
Thus, 140 people prefer tea.
2. If the sum of the angles in a triangle is 180 degrees, what is the measure of the third angle if the other two are 60 and 70 degrees?
To find the third angle:
Third angle=180−(60+70)=180−130=50 degrees\text{Third angle} = 180 - (60 + 70) = 180 - 130 = 50 \text{ degrees}Third angle=180−(60+70)=180−130=50 degrees
Thus, the measure of the third angle is 50 degrees.
3. A man has twice as many apples as his friend. If together they have 36 apples, how many apples does each have?
Let the number of apples the friend has be xxx. Then, the man has 2x2x2x.
x+2x=36 ⟹ 3x=36 ⟹ x=12x + 2x = 36 \implies 3x = 36 \implies x = 12x+2x=36⟹3x=36⟹x=12
2x=2×12=242x = 2 \times 12 = 242x=2×12=24
Thus, the friend has 12 apples, and the man has 24 apples.
4. If the cost of 3 notebooks is $9, what is the cost of 8 notebooks?
To find the cost of one notebook:
Cost per notebook=93=3 dollars\text{Cost per notebook} = \frac{9}{3} = 3 \text{ dollars}Cost per notebook=39=3 dollars
Cost of 8 notebooks=8×3=24 dollars\text{Cost of 8 notebooks} = 8 \times 3 = 24 \text{ dollars}Cost of 8 notebooks=8×3=24 dollars
Thus, the cost of 8 notebooks is $24.
5. If a sum of money is divided among A, B, and C in the ratio 3:4:5, what fraction does B receive if the total amount is $120?
The total parts in the ratio:
3+4+5=12 parts3 + 4 + 5 = 12 \text{ parts}3+4+5=12 parts
B’s share=412×120=13×120=40 dollars\text{B's share} = \frac{4}{12} \times 120 = \frac{1}{3} \times 120 = 40 \text{ dollars}B’s share=124×120=31×120=40 dollars
Thus, B receives $40.
6. If a rectangular prism has a length of 10 cm, a width of 5 cm, and a height of 2 cm, what is its volume?
Volume of a rectangular prism is calculated as:
Volume=Length×Width×Height=10×5×2=100 cubic cm\text{Volume} = \text{Length} \times \text{Width} \times \text{Height} = 10 \times 5 \times 2 = 100 \text{ cubic cm}Volume=Length×Width×Height=10×5×2=100 cubic cm
Thus, the volume is 100 cubic cm.
7. If an investment of $1,000 earns 5% interest per year, how much will it be worth after 3 years?
To find the total amount after 3 years with simple interest:
Interest=Principal×Rate×Time=1000×0.05×3=150\text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time} = 1000 \times 0.05 \times 3 = 150Interest=Principal×Rate×Time=1000×0.05×3=150
Total amount=1000+150=1150 dollars\text{Total amount} = 1000 + 150 = 1150 \text{ dollars}Total amount=1000+150=1150 dollars
Thus, the investment will be worth $1,150 after 3 years.
8. In a class of 50 students, 30 passed Math, 25 passed Science, and 15 passed both. How many students passed at least one subject?
To find the number of students who passed at least one subject, use the formula:
Passed at least one=Passed Math+Passed Science−Passed both\text{Passed at least one} = \text{Passed Math} + \text{Passed Science} - \text{Passed both}Passed at least one=Passed Math+Passed Science−Passed both
Passed at least one=30+25−15=40\text{Passed at least one} = 30 + 25 - 15 = 40Passed at least one=30+25−15=40
Thus, 40 students passed at least one subject.
9. A factory produces 500 toys in 8 hours. How many toys can it produce in 12 hours at the same rate?
First, find the rate of production per hour:
Rate=5008=62.5 toys per hour\text{Rate} = \frac{500}{8} = 62.5 \text{ toys per hour}Rate=8500=62.5 toys per hour
- Calculation for 12 hours:
Toys produced in 12 hours=62.5×12=750 toys\text{Toys produced in 12 hours} = 62.5 \times 12 = 750 \text{ toys}Toys produced in 12 hours=62.5×12=750 toys
Thus, the factory can produce 750 toys in 12 hours.
10. If a box contains 20 balls: 10 red, 5 blue, and 5 green, what is the probability of randomly selecting a blue ball?
Total number of balls in the box:
20 balls=10 red+5 blue+5 green20 \text{ balls} = 10 \text{ red} + 5 \text{ blue} + 5 \text{ green}20 balls=10 red+5 blue+5 green
- Probability of selecting a blue ball:
P(Blue)=520=14P(\text{Blue}) = \frac{5}{20} = \frac{1}{4}P(Blue)=205=41
Thus, the probability of randomly selecting a blue ball is 14\frac{1}{4}41.
4o mini
11. If a car travels 150 miles on 5 gallons of gas, what is its mileage in miles per gallon?
To calculate mileage (miles per gallon), use the formula:
Mileage=DistanceGallons=150 miles5 gallons=30 miles per gallon\text{Mileage} = \frac{\text{Distance}}{\text{Gallons}} = \frac{150 \text{ miles}}{5 \text{ gallons}} = 30 \text{ miles per gallon}Mileage=GallonsDistance=5 gallons150 miles=30 miles per gallon
Thus, the car's mileage is 30 miles per gallon.
12. In a family of six members, the average age is 30 years. What is the total age of all family members?
To find the total age, multiply the average age by the number of family members:
Total age=Average age×Number of members=30×6=180 years\text{Total age} = \text{Average age} \times \text{Number of members} = 30 \times 6 = 180 \text{ years}Total age=Average age×Number of members=30×6=180 years
Thus, the total age of all family members is 180 years.
13. A bag contains 12 white balls and 8 black balls. What is the probability of picking a white ball?
Total number of balls in the bag:
12 white+8 black=20 balls12 \text{ white} + 8 \text{ black} = 20 \text{ balls}12 white+8 black=20 balls
- Probability of selecting a white ball:
P(White)=1220=35P(\text{White}) = \frac{12}{20} = \frac{3}{5}P(White)=2012=53
Thus, the probability of picking a white ball is 35\frac{3}{5}53.
14. If the ratio of boys to girls in a class is 4:5 and there are 36 students in total, how many girls are there?
Let the number of boys be 4x4x4x and the number of girls be 5x5x5x.
4x+5x=36 ⟹ 9x=36 ⟹ x=44x + 5x = 36 \implies 9x = 36 \implies x = 44x+5x=36⟹9x=36⟹x=4
5x=5×4=205x = 5 \times 4 = 205x=5×4=20
Thus, there are 20 girls in the class.
15. If a shopkeeper sells an item for $120 after a 25% discount, what was the original price?
Let the original price be xxx. The selling price after a 25% discount is:
x−0.25x=0.75xx - 0.25x = 0.75xx−0.25x=0.75x
0.75x=120 ⟹ x=1200.75=1600.75x = 120 \implies x = \frac{120}{0.75} = 1600.75x=120⟹x=0.75120=160
Thus, the original price was $160.
16. If the sides of a square are doubled, what happens to the area?
The area of a square is given by:
Area=side2\text{Area} = \text{side}^2Area=side2
If the side length is doubled, the new area becomes:
New area=(2×side)2=4×side2\text{New area} = (2 \times \text{side})^2 = 4 \times \text{side}^2New area=(2×side)2=4×side2
The area increases by a factor of 4. Thus, the area increases by 4 times.
17. If a person can complete a task in 10 hours, how much of the task can they complete in 4 hours?
To find out how much of the task is completed in 4 hours, use the formula:
Work completed=TimeTotal time=410=0.4\text{Work completed} = \frac{\text{Time}}{\text{Total time}} = \frac{4}{10} = 0.4Work completed=Total timeTime=104=0.4
Thus, the person can complete 40% of the task in 4 hours.
18. If the median of the numbers 3, 7, 9, and xxx is 7, what can we say about the value of xxx?
To find the median:
- If x≤3x \leq 3x≤3: The ordered set is x,3,7,9x, 3, 7, 9x,3,7,9 → Median is 3+72=5\frac{3 + 7}{2} = 523+7=5 (not 7).
- If 3<x<73 < x < 73<x<7: The ordered set is 3,x,7,93, x, 7, 93,x,7,9 → Median is x+72\frac{x + 7}{2}2x+7 = 7 → x=7x = 7x=7 (valid).
- If x=7x = 7x=7: The ordered set is 3,7,7,93, 7, 7, 93,7,7,9 → Median is 7+72=7\frac{7 + 7}{2} = 727+7=7 (valid).
- If x>7x > 7x>7: The ordered set is 3,7,9,x3, 7, 9, x3,7,9,x → Median is 7+92=8\frac{7 + 9}{2} = 827+9=8 (not 7).
Thus, xxx must be equal to 7.
19. A computer can process 1000 operations in a second. How many operations can it process in 5 minutes?
To find the total operations in 5 minutes:
- Convert minutes to seconds:
5 minutes=5×60=300 seconds5 \text{ minutes} = 5 \times 60 = 300 \text{ seconds}5 minutes=5×60=300 seconds
- Calculate total operations:
Total operations=1000×300=300,000\text{Total operations} = 1000 \times 300 = 300,000Total operations=1000×300=300,000
Thus, the computer can process 300,000 operations in 5 minutes.
20. If a triangle has sides of lengths 5, 12, and 13, is it a right triangle?
To determine if it’s a right triangle, use the Pythagorean theorem:
a2+b2=c2a^2 + b^2 = c^2a2+b2=c2
where ccc is the longest side (13):
52+122=132 ⟹ 25+144=1695^2 + 12^2 = 13^2 \implies 25 + 144 = 16952+122=132⟹25+144=169
Since both sides are equal, the triangle is a right triangle.
21. If the probability of winning a game is 0.4, what is the probability of losing?
The probability of losing is the complement of the probability of winning. Therefore:
P(Losing)=1−P(Winning)=1−0.4=0.6P(\text{Losing}) = 1 - P(\text{Winning}) = 1 - 0.4 = 0.6P(Losing)=1−P(Winning)=1−0.4=0.6
Thus, the probability of losing is 0.6.
22. If the ages of two siblings are in the ratio 3:4 and the sum of their ages is 28, how old is the older sibling?
Let the ages of the siblings be 3x3x3x and 4x4x4x. According to the given information:
3x+4x=28 ⟹ 7x=28 ⟹ x=43x + 4x = 28 \implies 7x = 28 \implies x = 43x+4x=28⟹7x=28⟹x=4
The ages are:
- Younger sibling: 3x=3×4=123x = 3 \times 4 = 123x=3×4=12
- Older sibling: 4x=4×4=164x = 4 \times 4 = 164x=4×4=16 Thus, the older sibling is 16 years old.
23. A train travels 300 miles at a speed of 75 mph. How long does it take to complete the journey?
To find the time taken, use the formula:
Time=DistanceSpeed=300 miles75 mph=4 hours\text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{300 \text{ miles}}{75 \text{ mph}} = 4 \text{ hours}Time=SpeedDistance=75 mph300 miles=4 hours
Thus, the journey takes 4 hours.
24. If a triangle has two angles measuring 50 and 60 degrees, what is the measure of the third angle?
The sum of the angles in a triangle is always 180 degrees. Therefore:
Third angle=180−(50+60)=180−110=70 degrees\text{Third angle} = 180 - (50 + 60) = 180 - 110 = 70 \text{ degrees}Third angle=180−(50+60)=180−110=70 degrees
Thus, the measure of the third angle is 70 degrees.
25. In a class, 60% of students passed Mathematics, and 50% passed Science. If 30% passed both, what percentage passed at least one subject?
Using the principle of inclusion-exclusion:
P(At least one)=P(Math)+P(Science)−P(Both)P(\text{At least one}) = P(\text{Math}) + P(\text{Science}) - P(\text{Both})P(At least one)=P(Math)+P(Science)−P(Both)
Substituting the values:
P(At least one)=60%+50%−30%=80%P(\text{At least one}) = 60\% + 50\% - 30\% = 80\%P(At least one)=60%+50%−30%=80%
Thus, 80% of the students passed at least one subject.
26. If the area of a circle is 78.5 square cm, what is its radius (use π=3.14\pi = 3.14π=3.14)?
The area AAA of a circle is given by:
A=πr2A = \pi r^2A=πr2
Setting this equal to the given area:
78.5=3.14r278.5 = 3.14 r^278.5=3.14r2
Solving for r2r^2r2:
r2=78.53.14≈25 ⟹ r=25=5 cmr^2 = \frac{78.5}{3.14} \approx 25 \implies r = \sqrt{25} = 5 \text{ cm}r2=3.1478.5≈25⟹r=25=5 cm
Thus, the radius is 5 cm.
27. If a rectangular garden is 15 meters long and 10 meters wide, what is its area?
The area AAA of a rectangle is calculated as:
A=Length×Width=15 m×10 m=150 square metersA = \text{Length} \times \text{Width} = 15 \text{ m} \times 10 \text{ m} = 150 \text{ square meters}A=Length×Width=15 m×10 m=150 square meters
Thus, the area of the garden is 150 square meters.
28. If the product of three consecutive integers is 210, what are the integers?
Let the three consecutive integers be x−1x - 1x−1, xxx, and x+1x + 1x+1. Then:
(x−1)×x×(x+1)=210(x - 1) \times x \times (x + 1) = 210(x−1)×x×(x+1)=210
Testing x=6x = 6x=6:
5×6×7=2105 \times 6 \times 7 = 2105×6×7=210
Thus, the integers are 5, 6, and 7.
29. If a stock price increases from $50 to $75, what is the percentage increase?
The percentage increase is calculated using the formula:
Percentage Increase=New Price−Old PriceOld Price×100\text{Percentage Increase} = \frac{\text{New Price} - \text{Old Price}}{\text{Old Price}} \times 100Percentage Increase=Old PriceNew Price−Old Price×100
Substituting the values:
Percentage Increase=75−5050×100=2550×100=50%\text{Percentage Increase} = \frac{75 - 50}{50} \times 100 = \frac{25}{50} \times 100 = 50\%Percentage Increase=5075−50×100=5025×100=50%
Thus, the percentage increase is 50%.
30. In a box with 3 red, 4 green, and 5 blue marbles, what is the probability of picking a green marble?
Total number of marbles:
3+4+5=123 + 4 + 5 = 123+4+5=12
Probability of picking a green marble:
P(Green)=412=13P(\text{Green}) = \frac{4}{12} = \frac{1}{3}P(Green)=124=31
Thus, the probability of picking a green marble is 13\frac{1}{3}31.
31. If you flip a coin three times, what is the probability of getting exactly two heads?
The total number of outcomes when flipping a coin three times is 23=82^3 = 823=8. The combinations for getting exactly two heads (HHT, HTH, THH) can be calculated using the binomial coefficient:
Number of favorable outcomes=(32)=3\text{Number of favorable outcomes} = \binom{3}{2} = 3Number of favorable outcomes=(23)=3
Thus, the probability is:
P(Exactly 2 Heads)=38P(\text{Exactly 2 Heads}) = \frac{3}{8}P(Exactly 2 Heads)=83
Thus, the probability of getting exactly two heads is 38\frac{3}{8}83.
32. If a person earns $200 per day, how much will they earn in a 30-day month?
To find the total earnings, multiply the daily earnings by the number of days:
Total Earnings=200 per day×30 days=6000\text{Total Earnings} = 200 \text{ per day} \times 30 \text{ days} = 6000Total Earnings=200 per day×30 days=6000
Thus, the person will earn $6000 in a 30-day month.
33. If a bag contains 4 apples, 6 bananas, and 10 oranges, what is the probability of randomly picking an apple?
Total number of fruits:
4 apples+6 bananas+10 oranges=204 \text{ apples} + 6 \text{ bananas} + 10 \text{ oranges} = 204 apples+6 bananas+10 oranges=20
Probability of picking an apple:
P(Apple)=420=15P(\text{Apple}) = \frac{4}{20} = \frac{1}{5}P(Apple)=204=51
Thus, the probability of randomly picking an apple is 15\frac{1}{5}51.
34. If the ages of three friends are in the ratio 2:3:4 and the total age is 72, what is the age of the oldest friend?
Let the ages be 2x2x2x, 3x3x3x, and 4x4x4x. According to the given information:
2x+3x+4x=72 ⟹ 9x=72 ⟹ x=82x + 3x + 4x = 72 \implies 9x = 72 \implies x = 82x+3x+4x=72⟹9x=72⟹x=8
- Ages of the friends:some text
- First friend: 2x=162x = 162x=16
- Second friend: 3x=243x = 243x=24
- Oldest friend: 4x=324x = 324x=32 Thus, the age of the oldest friend is 32 years.
35. If the sum of three consecutive even numbers is 54, what are the numbers?
Let the consecutive even numbers be xxx, x+2x + 2x+2, and x+4x + 4x+4. Then:
x+(x+2)+(x+4)=54x + (x + 2) + (x + 4) = 54x+(x+2)+(x+4)=54
Simplifying:
3x+6=54 ⟹ 3x=48 ⟹ x=163x + 6 = 54 \implies 3x = 48 \implies x = 163x+6=54⟹3x=48⟹x=16
The numbers are:
- First: 161616
- Second: 181818
- Third: 202020 Thus, the consecutive even numbers are 16, 18, and 20.
36. If a train leaves a station at 10 AM and travels at 60 mph, what time will it arrive at a destination 180 miles away?
To find the travel time:
Time=DistanceSpeed=180 miles60 mph=3 hours\text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{180 \text{ miles}}{60 \text{ mph}} = 3 \text{ hours}Time=SpeedDistance=60 mph180 miles=3 hours
Departure time: 10 AM + 3 hours = 1 PM. Thus, the train will arrive at 1 PM.
37. If a company produces 200 units of a product and sells each for $50, what is the total revenue?
Total revenue is calculated as:
Total Revenue=Price per Unit×Number of Units=50×200=10,000\text{Total Revenue} = \text{Price per Unit} \times \text{Number of Units} = 50 \times 200 = 10,000Total Revenue=Price per Unit×Number of Units=50×200=10,000
Thus, the total revenue is $10,000.
38. If the circumference of a circle is 31.4 cm, what is its diameter (use π=3.14\pi = 3.14π=3.14)?
The circumference CCC of a circle is given by:
C=π×dC = \pi \times dC=π×d
Setting this equal to the given circumference:
31.4=3.14×d31.4 = 3.14 \times d31.4=3.14×d
Solving for ddd:
d=31.43.14=10 cmd = \frac{31.4}{3.14} = 10 \text{ cm}d=3.1431.4=10 cm
Thus, the diameter is 10 cm.
39. If a student scored 85 out of 100 in one test and 90 out of 100 in another, what is their average score?
To calculate the average score:
Average=Score 1+Score 2Number of Tests=85+902=1752=87.5\text{Average} = \frac{\text{Score 1} + \text{Score 2}}{\text{Number of Tests}} = \frac{85 + 90}{2} = \frac{175}{2} = 87.5Average=Number of TestsScore 1+Score 2=285+90=2175=87.5
Thus, the average score is 87.5.
40. If an item costs $80 after a 20% increase, what was its original price?
Let the original price be xxx. The price after a 20% increase can be expressed as:
x+0.2x=1.2xx + 0.2x = 1.2xx+0.2x=1.2x
Setting this equal to the selling price:
1.2x=80 ⟹ x=801.2=80012≈66.671.2x = 80 \implies x = \frac{80}{1.2} = \frac{800}{12} \approx 66.671.2x=80⟹x=1.280=12800≈66.67
Thus, the original price was approximately $66.67.
