ISEE Upper Level Quiz: Ratios And Comparisons

20 questions · exam conditions

0:00

Ratios And ComparisonsQuestion 1 of 20

A factory produces red, blue, and green widgets in the ratio $6:4:5$ . If production increases such that red widget output doubles, blue widget output increases by 50%, and green widget output remains constant, what is the new ratio of red to blue to green widgets?

12:6:5

8:3:5

6:2:5

12:8:5

← Back to quizzes

ISEE Upper Level Quiz

ISEE Upper Level Quiz: Ratios And Comparisons

Practice Ratios And Comparisons in ISEE Upper Level with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Ratios And Comparisons, giving you a quick way to practice the rules, question types, and explanations that matter most for ISEE Upper Level.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

$12:6:5$ (correct answer)
$8:3:5$
$6:2:5$
$12:8:5$

Explanation: Ratio problems involving changes require you to apply percentage increases to each component separately, then express the results in simplest form. Start with the original ratio of red:blue:green widgets at

6:4:5

. Now apply each change:

Red widgets double: $6 \times 2 = 12$
Blue widgets increase by 50%: $4 \times 1.5 = 6$
Green widgets remain constant: $5$

This gives you the new ratio

12:6:5

, which is already in simplest form since 12, 6, and 5 share no common factors other than 1. Looking at the wrong answers: Choice B (

8:3:5

) appears to come from incorrectly calculating the blue increase as 25% instead of 50%, giving

4 \times 1.25 = 5

, then somehow arriving at different values entirely. Choice C (

6:2:5

) suggests someone misunderstood "doubles" as "increases by 100%" but applied it incorrectly, and severely miscalculated the blue increase. Choice D (

12:8:5

) correctly doubles the red widgets but incorrectly doubles the blue widgets too, instead of applying the 50% increase. The correct answer is A:

12:6:5

. Remember that "increases by X%" means you multiply by

(1 + \frac{X}{100})

, while "doubles" means you multiply by 2. Always double-check that you're applying the right operation to each component, and verify your final ratio is in simplest form by checking for common factors.

Question 2

A concrete mixture contains cement, sand, and gravel in the ratio $1:3:4$ . If a construction project uses 24 cubic yards of this mixture and requires an additional 6 cubic yards of sand to be added separately, what is the ratio of sand to the total volume of materials?

$15:30$
$1:2$ (correct answer)
$9:30$
$3:10$

Explanation: When you encounter ratio problems that involve adding materials, you need to carefully track both the original components and any additions to find the new total. Start by finding how much of each material is in the original 24 cubic yards of concrete. The ratio

1:3:4

means for every 8 parts total (

1+3+4=8

), cement gets 1 part, sand gets 3 parts, and gravel gets 4 parts. So the original mixture contains:

\frac{3}{8} \times 24 = 9

cubic yards of sand, along with 3 cubic yards of cement and 12 cubic yards of gravel. After adding 6 cubic yards of sand separately, you have

9 + 6 = 15

cubic yards of sand total. The total volume of all materials is now

24 + 6 = 30

cubic yards. Therefore, the ratio of sand to total volume is

15:30

, which simplifies to

1:2

. Choice A (

15:30

) represents the unsimplified form of the correct ratio, but ratios should always be reduced to lowest terms. Choice C (

9:30

) incorrectly uses only the original sand amount, forgetting to include the 6 cubic yards added separately. Choice D (

3:10

) appears to use the original ratio of sand to total concrete (

3:8

) but incorrectly adjusted. Always remember to simplify ratios to their lowest terms, and when materials are added to an existing mixture, update both the component amounts and the total volume before calculating your final ratio.

Question 3

In a chess tournament, the ratio of wins to losses to draws for Player A is $4:1:2$ . Player A has played 21 games total. If Player A wins the next 3 games, what will be the new ratio of wins to total games played?

$15:24$
$5:8$ (correct answer)
$12:21$
$4:7$

Explanation: Ratio problems require you to work with parts and wholes systematically. When you see a ratio like

4:1:2

, these numbers represent relative parts, not actual quantities. First, find the actual number of wins, losses, and draws. The ratio

4:1:2

means for every 7 total parts (

4+1+2=7

), Player A has 4 wins, 1 loss, and 2 draws. Since Player A played 21 games total, each part represents

21 \div 7 = 3

games. Therefore: wins =

4 \times 3 = 12

, losses =

1 \times 3 = 3

, draws =

2 \times 3 = 6

. After winning the next 3 games, Player A will have

12 + 3 = 15

wins out of

21 + 3 = 24

total games. The ratio of wins to total games is

15:24

. To simplify, divide both numbers by their greatest common factor of 3:

15:24 = 5:8

. Choice A (

15:24

) represents the unsimplified ratio - a common trap since this is the correct relationship but not in simplest form. Choice C (

12:21

) uses the original wins and games before the additional 3 victories. Choice D (

4:7

) incorrectly applies the original win ratio (4 parts out of 7 total parts) without accounting for the changed situation. The correct answer is B. Strategy tip: Always check whether ratios need to be simplified on the ISEE. Calculate the actual quantities first, then form your new ratio, and finally reduce to lowest terms by finding the greatest common factor.

Question 4

Two pipes fill a tank in the ratio $3:5$ regarding their flow rates. If the faster pipe can fill the tank alone in 15 hours, how long would it take both pipes together to fill $\frac{2}{3}$ of the tank?

6.25 hours together (correct answer)
7.5 hours together
5.0 hours together
10.0 hours together

Explanation: When you encounter pipe flow problems, you're dealing with rates of work. The key insight is that flow rates are inversely proportional to time—faster pipes take less time to fill the same volume. Since the pipes have flow rates in the ratio

3:5

, and the faster pipe (ratio 5) fills the tank in 15 hours, you can find the slower pipe's time. If the faster pipe has rate

\frac{1}{15}

tanks per hour, then the slower pipe has rate

\frac{3}{5} \times \frac{1}{15} = \frac{1}{25}

tanks per hour, meaning it takes 25 hours alone. When working together, their combined rate is

\frac{1}{15} + \frac{1}{25}

. Finding a common denominator:

\frac{5}{75} + \frac{3}{75} = \frac{8}{75}

tanks per hour. To fill

\frac{2}{3}

of the tank at rate

\frac{8}{75}

: Time =

\frac{\frac{2}{3}}{\frac{8}{75}} = \frac{2}{3} \times \frac{75}{8} = \frac{150}{24} = 6.25

hours. Choice A (6.25 hours) is correct. Choice B (7.5 hours) likely comes from incorrectly calculating the time to fill the entire tank together, then making an error with the

\frac{2}{3}

factor. Choice C (5.0 hours) might result from using only the faster pipe's rate for the entire calculation. Choice D (10.0 hours) could come from incorrectly finding

\frac{2}{3}

of the slower pipe's individual time. Remember: always add the rates when pipes work together, then divide the desired volume by this combined rate.

Question 5

A rectangular garden has length to width in the ratio $5:3$ . If the perimeter is 64 feet and the garden is expanded by increasing the length by 20% and the width by 10%, what is the new ratio of length to width?

$6:3.3$
$60:33$
$20:11$ (correct answer)
$24:13$

Explanation: When you encounter ratio and percentage problems, you need to work systematically through the original dimensions, then apply the changes to find the new ratio. Start with the original ratio of length to width as

5:3

. This means length =

5x

and width =

3x

for some value

x

. Since perimeter =

2(\text{length} + \text{width}) = 64

, you have

2(5x + 3x) = 64

, so

16x = 64

and

x = 4

. Therefore, the original length is

5(4) = 20

feet and width is

3(4) = 12

feet. After expansion, the new length is

20 \times 1.20 = 24

feet, and the new width is

12 \times 1.10 = 13.2

feet. The new ratio is

24:13.2

. To simplify this to whole numbers, multiply both parts by 10 to get

240:132

, then divide by their greatest common factor of 12 to get

20:11

. Choice A (

6:3.3

) represents the original ratio multiplied by the percentage increases (

5 \times 1.2 : 3 \times 1.1

) but doesn't account for finding the actual dimensions first. Choice B (

60:33

) makes the same error but multiplies by 10. Choice D (

24:13

) correctly finds the new length but incorrectly rounds the width to 13 instead of 13.2. Remember: when dealing with ratios and percentages, always find the actual values first, apply the changes, then simplify the resulting ratio. Don't apply percentage changes directly to the ratio parts without considering the underlying dimensions.

Question 6

In a survey, the ratio of people who prefer coffee to tea to juice is $8:5:2$ . If 45 people prefer tea, and 18 people change their preference from coffee to juice, what is the new ratio of people preferring coffee to people preferring juice?

$54:36$
$3:2$ (correct answer)
$2:1$
$9:6$

Explanation: When you encounter ratio problems with changing quantities, you need to first establish the actual numbers from the given ratio, then apply the changes to find the new relationship. Given the ratio

8:5:2

for coffee:tea:juice preferences and knowing 45 people prefer tea, you can find the scaling factor. Since tea represents 5 parts and equals 45 people, each part represents

45 ÷ 5 = 9

people. This means initially there are

8 × 9 = 72

coffee drinkers and

2 × 9 = 18

juice drinkers. After 18 people switch from coffee to juice, the new numbers become: coffee =

72 - 18 = 54

people, and juice =

18 + 18 = 36

people. The ratio of coffee to juice is

54:36

, which simplifies to

3:2

by dividing both terms by 18. Looking at the wrong answers: Choice A (

54:36

) gives the actual numbers but isn't simplified to lowest terms as ratios typically should be. Choice C (

2:1

) incorrectly suggests coffee drinkers are double the juice drinkers, but

54:36

reduces to

3:2

, not

2:1

. Choice D (

9:6

) also simplifies to

3:2

, but uses different terms than the fully reduced form. The key strategy for ratio problems is to always find the actual quantities first using the given information, apply any changes, then express your final answer as a simplified ratio. Don't forget to reduce to lowest terms unless the problem specifically asks for actual numbers.

Question 7

The ratio of red marbles to blue marbles in a jar is $7:3$ . If the total number of marbles is 150, and 20 green marbles are added to the jar, what will be the new ratio of red marbles to the total number of marbles?

$7:17$
$105:170$
$7:13$
$21:34$ (correct answer)

Explanation: When you encounter ratio problems with changing quantities, you need to work systematically through the original amounts, then recalculate after the change. Start with the original ratio of red to blue marbles:

7:3

. This means for every 7 red marbles, there are 3 blue marbles, making 10 total parts. Since there are 150 marbles total, each part represents

150 ÷ 10 = 15

marbles. Therefore, there are

7 × 15 = 105

red marbles and

3 × 15 = 45

blue marbles. After adding 20 green marbles, the total becomes

150 + 20 = 170

marbles. The number of red marbles stays the same at 105. So the new ratio of red marbles to total marbles is

105:170

. To simplify this ratio, find the greatest common divisor of 105 and 170. Both numbers are divisible by 5:

105 ÷ 5 = 21

and

170 ÷ 5 = 34

. Therefore, the simplified ratio is

21:34

, which is answer D. Let's examine the wrong answers: A)

7:17

incorrectly assumes the ratio parts stay the same after adding marbles. B)

105:170

gives the correct numbers but fails to simplify the ratio to lowest terms. C)

7:13

appears to use incorrect calculations for the total number of marbles. Always remember to simplify ratios to their lowest terms by finding the greatest common divisor. Also, when quantities change in ratio problems, recalculate the total carefully before forming your new ratio.

Question 8

A recipe calls for flour and sugar in the ratio $5:2$ . If Maria uses 15 cups of flour, how much sugar should she use to maintain the same ratio?

6 cups of sugar (correct answer)
7.5 cups of sugar
10 cups of sugar
37.5 cups of sugar

Explanation: When you see ratio problems, you're working with proportional relationships where quantities maintain a constant relationship to each other. The key is setting up a proportion to find the unknown value. The recipe uses flour and sugar in a

5:2

ratio, meaning for every 5 parts flour, you need 2 parts sugar. Since Maria uses 15 cups of flour, you need to find how many "parts" this represents:

15 ÷ 5 = 3

parts. If the flour is 3 times the base amount, the sugar must also be 3 times its base amount:

2 × 3 = 6

cups of sugar. You can also solve this using cross-multiplication:

\frac{5}{2} = \frac{15}{x}

, which gives you

5x = 30

, so

x = 6

. Looking at the wrong answers: Choice B (7.5 cups) comes from incorrectly thinking you multiply the flour amount by the sugar ratio:

15 × \frac{2}{5} = 6

, but then making an arithmetic error. Choice C (10 cups) results from confusing the ratios and thinking it's

2:1

instead of

5:2

. Choice D (37.5 cups) comes from mistakenly multiplying

15 × 2.5

(which would be

15 × \frac{5}{2}

), completely reversing the relationship. The answer is A: 6 cups of sugar. Strategy tip: In ratio problems, always identify what one "part" equals by dividing the known quantity by its ratio number, then multiply by the unknown quantity's ratio number. This two-step approach prevents most ratio errors.

Question 9

In a mixture, the ratio of water to orange juice is $2:3$ . If 10 liters of water and 5 liters of orange juice are added to 40 liters of this mixture, what is the new ratio of water to orange juice?

$1:1$
$26:29$ (correct answer)
$3:4$
$22:23$

Explanation: When you encounter ratio problems involving mixtures, the key is to track each component separately through all changes, then form the new ratio at the end. Start by finding how much water and orange juice are in the original 40-liter mixture. With a

2:3

ratio of water to orange juice, the mixture contains

\frac{2}{2+3} = \frac{2}{5}

water and

\frac{3}{5}

orange juice. So there are

40 \times \frac{2}{5} = 16

liters of water and

40 \times \frac{3}{5} = 24

liters of orange juice initially. After adding 10 liters of water and 5 liters of orange juice, you have:

Water: $16 + 10 = 26$ liters
Orange juice: $24 + 5 = 29$ liters

The new ratio is

26:29

, which is answer choice B. Let's examine why the other answers are incorrect. Choice A (

1:1

) would require equal amounts of water and orange juice, but we have 26 and 29 liters respectively. Choice C (

3:4

) might tempt you if you mistakenly thought the original ratio would be preserved after adding equal proportions, but we're adding different amounts (10 vs 5 liters). Choice D (

22:23

) could result from calculation errors, perhaps incorrectly determining the original mixture contents. Remember: in mixture problems, always break down the original mixture into its components first, then add the new quantities to each component separately. Don't try to work with ratios directly when adding different amounts to each part.

Question 10

Two numbers are in the ratio $5:7$ . If the sum of the numbers is 96, what is the ratio of the smaller number to the difference between the larger and smaller numbers?

$5:2$ (correct answer)
$40:16$
$2:5$
$5:12$

Explanation: When you encounter ratio problems with given sums, set up the problem using variables that maintain the ratio relationship. Since the numbers are in the ratio

5:7

, you can represent them as

5x

and

7x

for some value

x

. Given that their sum is 96, you have:

5x + 7x = 96

, which simplifies to

12x = 96

, so

x = 8

. Therefore, the two numbers are

5(8) = 40

and

7(8) = 56

. The question asks for the ratio of the smaller number to the difference between the larger and smaller numbers. The smaller number is 40, and the difference is

56 - 40 = 16

. So the ratio is

40:16

, which simplifies to

5:2

by dividing both terms by 8. Looking at the wrong answers: Choice B (

40:16

) represents the unsimplified form of the correct ratio—always reduce ratios to lowest terms. Choice C (

2:5

) reverses the correct ratio, giving you the difference to the smaller number instead of smaller number to difference. Choice D (

5:12

) incorrectly uses the sum of both numbers (which relates to the

12x

from our equation) instead of their difference. Strategy tip: In multi-step ratio problems, work systematically: first find the actual numbers using the given constraint, then calculate what the question specifically asks for. Always check that your final answer addresses exactly what was requested, and remember to simplify ratios to lowest terms.

Question 11

In a parking lot, the ratio of cars to motorcycles to bicycles is $12:3:2$ . If there are 34 total vehicles and each bicycle is replaced with a car, what will be the new ratio of cars to motorcycles?

$7:1$
$14:3$ (correct answer)
$26:6$
$8:2$

Explanation: When you encounter ratio problems with multiple changes, break them down step by step. First, determine the actual quantities, then apply the changes, and finally calculate the new ratio. Start by finding how many of each vehicle type exists initially. The ratio

12:3:2

means for every 17 parts total (

12+3+2=17

), there are 34 vehicles. So each part equals

34÷17=2

vehicles. This gives us: 12 parts × 2 = 24 cars, 3 parts × 2 = 6 motorcycles, and 2 parts × 2 = 4 bicycles. Next, apply the change: each bicycle becomes a car. The 4 bicycles are replaced with 4 cars, giving us

24+4=28

cars and still 6 motorcycles. The new ratio is

28:6

, which simplifies to

14:3

by dividing both terms by 2. Looking at the wrong answers: Choice A (

7:1

) incorrectly reduces

14:3

further, perhaps by mistakenly dividing by different numbers. Choice C (

26:6

) likely comes from adding only 2 cars instead of 4, missing that all 4 bicycles become cars. Choice D (

8:2

) appears to be a random simplification that doesn't match the actual numbers. Remember that in ratio problems involving changes, always work with actual quantities first rather than trying to manipulate the ratios directly. Convert ratios to real numbers, make the specified changes, then create your new ratio and simplify if possible.

Question 12

Two machines produce widgets in the ratio $5:3$ . If Machine A produces 15 widgets per hour, what is the ratio of Machine B's hourly production to the total hourly production of both machines?

$9:24$
$3:8$ (correct answer)
$1:3$
$9:15$

Explanation: When you encounter ratio problems involving multiple machines or workers, focus on finding the actual production rates first, then use those to calculate any requested ratios. Given that the machines produce widgets in a

5:3

ratio and Machine A produces 15 widgets per hour, you can find Machine B's rate. Since Machine A corresponds to the "5" part of the ratio, each ratio unit represents

15 ÷ 5 = 3

widgets per hour. Therefore, Machine B produces

3 × 3 = 9

widgets per hour. The total hourly production is

15 + 9 = 24

widgets. The ratio of Machine B's production to total production is

9:24

, which simplifies to

3:8

by dividing both terms by 3. Looking at the wrong answers: Choice A gives

9:24

, which is the unsimplified form of the correct ratio—always simplify ratios to lowest terms. Choice C shows

1:3

, which incorrectly suggests Machine B produces only one-third of the total, when it actually produces three-eighths. Choice D presents

9:15

, which compares Machine B's production to only Machine A's production rather than the total production of both machines. Strategy tip: In ratio problems, always identify what each number in the original ratio represents in real units, calculate the actual values, then build your final ratio step by step. Watch out for answer choices that give correct intermediate calculations but don't answer the specific question asked.

Question 13

The ages of three siblings are in the ratio $2:3:5$ . In 4 years, the sum of their ages will be 46. What is the current ratio of the youngest sibling's age to the oldest sibling's age?

$6:15$
$2:5$ (correct answer)
$10:25$
$4:10$

Explanation: When you encounter ratio problems with age changes over time, the key insight is that ratios change as people age, but you can use the given information to find the actual current ages first. Since the current ages are in the ratio

2:3:5

, you can represent them as

2x

3x

, and

5x

for some value

x

. In 4 years, these ages become

(2x+4)

(3x+4)

, and

(5x+4)

. Their sum will be 46, so:

(2x+4) + (3x+4) + (5x+4) = 46

10x + 12 = 46

10x = 34

x = 3.4

Therefore, the current ages are

2(3.4) = 6.8

3(3.4) = 10.2

, and

5(3.4) = 17

years old. The ratio of youngest to oldest is

6.8:17

, which simplifies to

2:5

. Choice A (

6:15

) incorrectly assumes

x = 3

and creates a ratio that doesn't match the original

2:5

relationship. Choice C (

10:25

) appears to multiply the correct ratio by 5, perhaps confusing the sum with individual terms. Choice D (

4:10

) doubles the original ratio values, possibly from incorrectly adding the 4-year age increase to the ratio itself rather than solving for actual ages first. The correct answer is B. Remember: in ratio problems involving changes over time, always solve for the actual values first, then create the new ratio from those concrete numbers. Don't try to manipulate the ratios directly.

Question 14

A theater budget is sets:costumes $=7:5$ ; if costumes increase $50\%$ , what is the new ratio?

7:7.5 (correct answer)
7:10
10.5:5
5:7

Explanation: This question tests ISEE Upper Level quantitative reasoning skills, specifically calculating new ratios when one quantity changes by a percentage. The theater budget starts with sets:costumes = 7:5, and when costumes increase by 50%, the costume budget becomes 5 × 1.5 = 7.5, while sets remain at 7. The new ratio is therefore 7:7.5, which can also be expressed as 14:15 when both terms are multiplied by 2 to eliminate the decimal. Choice A is correct because it accurately represents the new ratio with sets at 7 and costumes at 7.5. Choice B incorrectly calculates the 50% increase, choice C reverses and miscalculates the ratio, and choice D shows the original ratio reversed. To master this concept, students should practice calculating percentage increases first, then form the new ratio, being comfortable with decimal values in ratios when appropriate.

Question 15

A school's student-to-teacher ratio is $25:1$ . If the school has 450 students, and 50 students transfer out while 2 new teachers are hired, what is the new student-to-teacher ratio?

$20:1$ (correct answer)
$400:20$
$19:1$
$25:1$

Explanation: Ratio problems require you to track how changes affect both parts of the relationship. When you see a ratio like

25:1

, this means there are 25 students for every 1 teacher. Start by finding the original number of teachers. If the ratio is

25:1

and there are 450 students, then

\frac{450}{25} = 18

teachers originally. Now apply the changes: 50 students transfer out, leaving

450 - 50 = 400

students. Two new teachers are hired, giving us

18 + 2 = 20

teachers. The new ratio is

400:20

, which simplifies to

20:1

by dividing both numbers by 20. Looking at the wrong answers: Choice B (

400:20

) shows the correct numbers but isn't simplified to lowest terms. While mathematically equivalent to

20:1

, ratios should always be expressed in simplest form. Choice C (

19:1

) likely comes from a calculation error, perhaps forgetting to add the 2 new teachers or making an arithmetic mistake. Choice D (

25:1

) ignores the changes entirely and keeps the original ratio. Remember that ratios must be simplified to their lowest terms, just like fractions. When working ratio problems, always identify what each number in the ratio represents, calculate the actual quantities, apply any changes systematically, then express your final answer in simplest form. Double-check by ensuring your ratio makes sense with the final numbers.

Question 16

In a photography club, the ratio of film cameras to digital cameras is $3:7$ . If there are 42 digital cameras, what is the ratio of film cameras to the total number of cameras?

$3:10$ (correct answer)
$18:60$
$3:7$
$18:42$

Explanation: When you encounter ratio problems, the key is to use the given information to find actual quantities, then create the new ratio being asked for. You're told the ratio of film to digital cameras is

3:7

, meaning for every 3 film cameras, there are 7 digital cameras. Since there are 42 digital cameras, you can set up a proportion: if 7 parts equal 42 cameras, then each part represents

42 ÷ 7 = 6

cameras. Therefore, there are

3 × 6 = 18

film cameras. The total number of cameras is

18 + 42 = 60

. The ratio of film cameras to total cameras is

18:60

, which simplifies to

3:10

(dividing both terms by 6). Looking at the wrong answers: Choice B (

18:60

) represents the same ratio as the correct answer but in unsimplified form - while mathematically equivalent, standardized tests typically expect simplified ratios. Choice C (

3:7

) is the original ratio of film to digital cameras, not film to total. Choice D (

18:42

) gives the ratio of film to digital cameras using the actual quantities, but again, this isn't what the question asks for. Remember that ratios can be written in different forms but should be simplified to lowest terms. Also, pay careful attention to what ratio is being requested - the question might give you one ratio (like part-to-part) but ask for a different one (like part-to-whole). Always identify exactly what quantities you need to compare in your final answer.

Question 17

Bus A travels 120 miles in 3 hours; Bus B travels 150 miles in 5 hours; what is $v_A:v_B$ ?

4:3 (correct answer)
3:4
40:30
8:5

Explanation: This question tests ISEE Upper Level quantitative reasoning skills, specifically calculating and comparing rates to form ratios. Ratios can compare any two quantities, including rates like speed, which is distance divided by time. Bus A travels 120 miles in 3 hours, giving a speed of 40 mph, while Bus B travels 150 miles in 5 hours, giving a speed of 30 mph. The ratio of speeds vA:vB is therefore 40:30, which simplifies to 4:3. Choice A is correct because it represents the simplified ratio of Bus A's speed to Bus B's speed. Choice B reverses the ratio, while choices C and D either don't simplify the ratio or use incorrect calculations. To help students master this skill, teach them to always calculate the individual rates first before forming the ratio, and to be careful about the order when writing ratios.

Question 18

A paint mixture contains red, blue, and yellow paint in the ratio $4:3:2$ . If the mixture contains 18 gallons of blue paint, how many total gallons of paint are in the mixture?

54 gallons total (correct answer)
45 gallons total
36 gallons total
27 gallons total

Explanation: Ratio problems test your ability to understand proportional relationships and scale them to find actual quantities. When you see a ratio like

4:3:2

, think of these as parts of a whole that maintain constant proportions. The ratio

4:3:2

means that for every 4 parts red paint, there are 3 parts blue paint and 2 parts yellow paint. Since you know there are 18 gallons of blue paint, you can find the value of one "part" by setting up a proportion. If 3 parts equals 18 gallons, then 1 part equals

18 \div 3 = 6

gallons. Now you can find each paint amount: red paint =

4 \times 6 = 24

gallons, blue paint =

3 \times 6 = 18

gallons (which matches the given information), and yellow paint =

2 \times 6 = 12

gallons. The total is

24 + 18 + 12 = 54

gallons. Looking at the wrong answers: B) 45 gallons likely comes from miscalculating one part as 5 gallons instead of 6. C) 36 gallons might result from only doubling the blue paint amount without properly using the ratio. D) 27 gallons could come from adding the ratio numbers (4+3+2=9) and multiplying by 3, which ignores the actual paint quantities. Study tip: Always verify your ratio work by checking that your calculated amount for the given quantity matches what's stated in the problem. This confirms you've found the correct value for one "part" before scaling up.

Question 19

A smoothie uses yogurt:fruit $=2:5$ ; which choice keeps the same ratio when doubled?

4:12
10:4
4:10 (correct answer)
6:15

Explanation: This question tests ISEE Upper Level quantitative reasoning skills, specifically understanding and applying ratios to compare quantities. Ratios are a way to compare two quantities by division, showing how many times one value contains or is contained within the other. The smoothie recipe uses yogurt to fruit in a 2:5 ratio, meaning for every 2 parts yogurt, there are 5 parts fruit. When doubling a recipe, both quantities must be multiplied by the same factor to maintain the ratio, so 2:5 becomes 4:10. Choice C is correct because 4:10 represents the doubled quantities while maintaining the same proportional relationship as 2:5. Choice A (4:12) incorrectly doubles the first term but more than doubles the second, while choices B and D don't maintain the original ratio structure. To master this skill, students should practice scaling ratios by multiplying both terms by the same factor and verify their work by simplifying the result back to the original ratio.

Question 20

A lab mixes Solution X:Solution Y $=3:2$ ; what is the ratio of Y to X?

3:2
2:3 (correct answer)
5:1
6:4

Explanation: This question tests ISEE Upper Level quantitative reasoning skills, specifically understanding how to reverse the order of terms in a ratio. Ratios express relationships between quantities in a specific order, and reversing this order creates the reciprocal ratio. The original ratio is Solution X:Solution Y = 3:2, meaning for every 3 parts of X, there are 2 parts of Y. When asked for the ratio of Y to X, we must reverse the order, giving us 2:3. Choice B is correct because it properly reverses the original ratio from X:Y to Y:X. Choice A incorrectly maintains the original order, while choices C and D use unrelated values. Students should be taught that when reversing a ratio, the first term becomes the second and vice versa, and to always pay careful attention to which quantity is being compared to which.