SSAT Middle Level Quiz: Proportional Relationships

20 questions · exam conditions

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Proportional RelationshipsQuestion 1 of 20

A backpack costs $60 and is 25% off. Using $p=0.75o$ , find the sale price.

$15

$35

$45

$75

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SSAT Middle Level Quiz

SSAT Middle Level Quiz: Proportional Relationships

Practice Proportional Relationships in SSAT Middle 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 Proportional Relationships, giving you a quick way to practice the rules, question types, and explanations that matter most for SSAT Middle 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

A backpack costs $60 and is 25% off. Using $p=0.75o$ , find the sale price.

$15
$35
$45 (correct answer)
$75

Explanation: This question tests middle school proportional reasoning skills, specifically solving proportional relationships using equations for discounts. Proportional reasoning involves understanding ratios and using them to find missing values in related quantities. For example, if an item is 25% off, you pay 75% of the original price. In this scenario, students use the equation p = 0.75o to determine the sale price by substituting o = 60. The correct answer works because it applies the principle of equal ratios, calculating 0.75 times 60 to get $45 accurately. A common distractor fails because it might result from calculating the discount amount only. Teaching strategies include practicing setting up and solving proportions from shopping contexts, emphasizing the importance of percentages and consistent ratios. Encourage students to cross-check their calculations and consider the practical implications of their results.

Question 2

At 48 miles per hour, a train travels $d=48t$ . How many miles in 1.75 hours?

49.75 miles
72 miles
84 miles (correct answer)
96 miles

Explanation: This question tests middle school proportional reasoning skills, specifically solving proportional relationships using equations for distance. Proportional reasoning involves understanding ratios and using them to find missing values in related quantities. For example, if a train travels at 48 mph, more time means proportionally more distance. In this scenario, students use the equation d = 48t to determine the distance by substituting t = 1.75. The correct answer works because it applies the principle of equal ratios, calculating 48 times 1.75 to get 84 miles accurately. A common distractor fails because it might result from decimal multiplication errors. Teaching strategies include practicing setting up and solving proportions from transportation contexts, emphasizing the importance of units and consistent ratios. Encourage students to cross-check their calculations and consider the practical implications of their results.

Question 3

A recipe for fruit punch calls for 3 cups of orange juice for every 5 cups of pineapple juice. If Maria wants to make punch using 12 cups of orange juice, how many total cups of punch will she make?

20 cups
32 cups (correct answer)
15 cups
17 cups
27 cups

Explanation: When you encounter ratio problems like this, you're working with proportional relationships where quantities must stay in the same relative amounts as the recipe scales up or down. The recipe calls for 3 cups orange juice to 5 cups pineapple juice. Since Maria wants to use 12 cups of orange juice, you need to find the scaling factor:

12 ÷ 3 = 4

. This means she's making 4 times the original recipe. If the orange juice increases by a factor of 4, the pineapple juice must also increase by the same factor to maintain the proper ratio:

5 × 4 = 20

cups of pineapple juice. The total punch will be:

12

cups orange juice

+ 20

cups pineapple juice

= 32

cups. This confirms answer B is correct. Looking at the wrong answers: Choice A (20 cups) represents just the amount of pineapple juice needed, not the total punch. Choice C (15 cups) might result from incorrectly adding the original recipe amounts (3 + 5) and then multiplying by 3, or from other calculation errors. Choice D (17 cups) could come from mistakenly adding 12 + 5, perhaps thinking the pineapple juice amount stays constant. The key strategy for ratio problems is to always find the scaling factor first, then apply it to all parts of the ratio consistently. Don't forget that the question asks for the total amount, so you'll need to add all ingredients together at the end.

Question 4

A car travels 240 miles in 4 hours at a constant speed. At this same rate, how long would it take to travel 420 miles?

6 hours
7 hours (correct answer)
8 hours
6.5 hours
7.5 hours

Explanation: When you encounter a problem involving constant speed and different distances, you're dealing with a rate problem that requires setting up a proportion or using the rate formula: distance = rate × time. First, find the car's constant speed by dividing the distance by time:

240 \text{ miles} ÷ 4 \text{ hours} = 60 \text{ mph}

. Now you can find how long it takes to travel 420 miles at this same rate:

420 \text{ miles} ÷ 60 \text{ mph} = 7 \text{ hours}

. This confirms answer B is correct. Alternatively, you could set up a proportion:

\frac{240 \text{ miles}}{4 \text{ hours}} = \frac{420 \text{ miles}}{x \text{ hours}}

. Cross-multiplying gives you

240x = 1680

, so

x = 7

. Looking at the wrong answers: Choice A (6 hours) would give you a speed of 70 mph, which is too fast based on the original conditions. Choice C (8 hours) would mean the car is traveling at only 52.5 mph, which is too slow. Choice D (6.5 hours) gives approximately 64.6 mph, again inconsistent with the established 60 mph rate. The key strategy for rate problems is to always establish the rate first, then use it consistently. Watch out for answer choices that seem reasonable but don't match the mathematical relationship. Double-check by working backwards: does your answer give you the original rate when you divide distance by time?

Question 5

In a scale drawing, 2 inches represents 15 feet. If a room is actually 24 feet long, what should its length be in the scale drawing?

3.0 inches
3.2 inches (correct answer)
3.6 inches
4.0 inches
4.2 inches

Explanation: Scale drawing problems test your ability to work with proportional relationships. When you see a question involving scale drawings, you're essentially solving a proportion where the ratio between drawing measurements and real measurements stays constant. Here, the scale tells you that 2 inches on the drawing represents 15 feet in reality. You need to find how many inches should represent 24 feet. Set up a proportion:

\frac{2 \text{ inches}}{15 \text{ feet}} = \frac{x \text{ inches}}{24 \text{ feet}}

Cross multiply:

2 \times 24 = 15 \times x

, which gives you

48 = 15x

. Solving for x:

x = \frac{48}{15} = 3.2

inches. Looking at the wrong answers: Choice A (3.0 inches) likely comes from rounding incorrectly or making an arithmetic error in the division. Choice C (3.6 inches) might result from accidentally using 18 instead of 15 in your calculation, or from setting up the proportion incorrectly. Choice D (4.0 inches) could come from using a simpler but wrong ratio like 1 inch to 6 feet, or from miscalculating

\frac{24}{6}

. The key strategy for scale drawing problems is to always set up your proportion carefully, keeping track of units. Write "inches" over "feet" on both sides of your equation, or "drawing" over "real" on both sides. This helps prevent the common mistake of flipping the ratio. Also, always double-check your arithmetic—these problems often involve division that doesn't come out to whole numbers.

Question 6

A machine fills 360 bottles in 45 minutes. At this rate, how many bottles can it fill in 2 hours and 15 minutes?

1,080 bottles (correct answer)
1,200 bottles
972 bottles
1,050 bottles
1,296 bottles

Explanation: This is a rate problem that tests your ability to work with proportional relationships and convert between different time units. First, you need to find the machine's rate. If it fills 360 bottles in 45 minutes, the rate is

\frac{360 \text{ bottles}}{45 \text{ minutes}} = 8 \text{ bottles per minute}

. Next, convert 2 hours and 15 minutes to minutes:

2 \times 60 + 15 = 135 \text{ minutes}

. Now multiply the rate by the time:

8 \text{ bottles/minute} \times 135 \text{ minutes} = 1,080 \text{ bottles}

. This confirms answer A is correct. Let's examine why the other answers are wrong. Answer B (1,200 bottles) likely comes from incorrectly calculating the rate as 10 bottles per minute instead of 8, then multiplying by 120 minutes (forgetting the extra 15 minutes). Answer C (972 bottles) might result from using the wrong conversion or making an arithmetic error in the multiplication. Answer D (1,050 bottles) could come from various calculation mistakes, perhaps using 131.25 minutes instead of 135 minutes. The key strategy for rate problems is to always establish the unit rate first (bottles per minute, miles per hour, etc.), then convert all time measurements to the same units before multiplying. Double-check your time conversions—mixing hours and minutes is a common source of errors on the SSAT. Setting up the problem as rate × time = total will keep you organized and help avoid calculation mistakes.

Question 7

A recipe calls for ingredients in the ratio flour:sugar:butter = 6:4:1. If Sarah uses 3 cups of butter, how many more cups of flour than sugar will she need?

6 cups more flour (correct answer)
8 cups more flour
4 cups more flour
12 cups more flour
10 cups more flour

Explanation: When you encounter ratio problems, you're working with proportional relationships where the actual quantities are multiples of the given ratio parts. The ratio flour:sugar:butter = 6:4:1 means for every 6 parts flour, there are 4 parts sugar and 1 part butter. Since Sarah uses 3 cups of butter, and butter represents 1 part in the ratio, each "part" equals 3 cups. This means:

Flour: $6 \times 3 = 18$ cups
Sugar: $4 \times 3 = 12$ cups
Butter: $1 \times 3 = 3$ cups

The difference between flour and sugar is

18 - 12 = 6

cups more flour. Looking at the wrong answers: Choice B (8 cups) might come from incorrectly thinking the difference in ratio parts (6-4=2) should be multiplied by 4 instead of recognizing that 2 parts × 3 cups per part = 6 cups. Choice C (4 cups) represents the raw difference in the ratio (6-4=2, then mistakenly doubled), missing the scaling factor entirely. Choice D (12 cups) could result from finding the amount of sugar (12 cups) rather than the difference between flour and sugar. The correct answer is A) 6 cups more flour. Strategy tip: In ratio problems, always identify what one "part" represents by finding which ingredient has a known quantity, then scale all other parts accordingly. Don't forget to answer what the question actually asks for—here it's the difference, not individual amounts.

Question 8

In a mixture, the ratio of water to orange juice is 3:5. If the mixture contains 24 ounces of water, what is the total volume of the mixture?

64.0 ounces total (correct answer)
40.0 ounces total
56.0 ounces total
48.0 ounces total
72.0 ounces total

Explanation: When you encounter ratio problems, you're working with proportional relationships between different parts of a whole. The key insight is that ratios tell you the relative amounts, and you can use given information to find the actual quantities. Here, the ratio of water to orange juice is 3:5, meaning for every 3 parts water, there are 5 parts orange juice. Since you know there are 24 ounces of water, you can set up a proportion. If 3 parts equals 24 ounces, then each part represents

24 ÷ 3 = 8

ounces. Since orange juice makes up 5 parts, it contains

5 × 8 = 40

ounces. The total mixture is water plus orange juice:

24 + 40 = 64

ounces. Looking at the wrong answers: Choice B (40.0 ounces) gives you only the amount of orange juice, not the total mixture. Choice C (56.0 ounces) might result from miscalculating the ratio or accidentally using 4:3 instead of 3:5. Choice D (48.0 ounces) could come from doubling the water amount instead of properly applying the ratio. The correct answer is A: 64.0 ounces total. Strategy tip: In ratio problems, always identify what each "part" represents by dividing the known quantity by its ratio number. Then multiply to find unknown quantities, and don't forget to add all parts together when asked for a total. Watch out for answer choices that give you only one component instead of the requested total.

Question 9

A gear with 24 teeth rotates 5 times while a connected gear with 40 teeth rotates 3 times. If the first gear makes 60 rotations, how many rotations will the second gear make?

36 rotations exactly (correct answer)
40 rotations exactly
45 rotations exactly
50 rotations exactly
32 rotations exactly

Explanation: When you encounter gear problems, you're working with inverse proportional relationships. As one gear gets bigger, it rotates slower, and the ratio of their teeth determines exactly how their rotation speeds relate. The key insight is that connected gears have a constant relationship: the product of teeth times rotations stays equal. So if the first gear has 24 teeth and rotates 5 times, that's

24 \times 5 = 120

"tooth-rotations." The second gear with 40 teeth rotating 3 times gives

40 \times 3 = 120

tooth-rotations. This confirms they're properly connected. Now you can set up the proportion. If the first gear makes 60 rotations, you need:

24 \times 60 = 40 \times x

, where x is the second gear's rotations. This gives you

1440 = 40x

, so

x = 36

rotations. Looking at the wrong answers: B) 40 rotations comes from mistakenly thinking the second gear rotates as many times as it has teeth, ignoring the gear relationship entirely. C) 45 rotations might result from incorrectly averaging the given numbers or using a flawed proportion. D) 50 rotations could come from mixing up the gear ratios or applying the relationship backwards. Strategy tip: In gear problems, always remember that larger gears rotate fewer times. Set up your proportion as

\text{teeth}_1 \times \text{rotations}_1 = \text{teeth}_2 \times \text{rotations}_2

. This "conservation of tooth-rotations" approach works every time and helps you avoid the common trap of thinking bigger gears rotate more.

Question 10

The cost of buying notebooks is directly proportional to the number purchased. If 8 notebooks cost $12, what is the cost of 15 notebooks?

$22.50 total cost (correct answer)
$18.00 total cost
$24.00 total cost
$20.00 total cost
$25.50 total cost

Explanation: When you see that one quantity is "directly proportional" to another, it means they have a constant ratio. As one increases, the other increases at the same rate. This is a fundamental relationship in many real-world situations like unit pricing. To solve this, first find the cost per notebook:

\frac{\$12}{8 \text{ notebooks}} = \$1.50 \text{ per notebook}

. Since the cost is directly proportional to the number purchased, each notebook costs the same amount. For 15 notebooks:

15 \times \$1.50 = \$22.50

. You can also set up a proportion:

\frac{8 \text{ notebooks}}{\$12} = \frac{15 \text{ notebooks}}{x}

. Cross-multiplying:

8x = 12 \times 15 = 180

, so

x = \$22.50

. Answer A (

22.50) is correct using either method. Answer B (

18.00) represents a common error where students might incorrectly calculate

15 ÷ 8 × 12 = 22.5

, then round down or make an arithmetic mistake. Answer C ($24.00) could result from incorrectly using $15 × $1.60 per notebook, possibly from rounding

12 ÷ 8 incorrectly. Answer D (

20.00) might come from approximating too early in the calculation or using faulty mental math. Strategy tip: For direct proportion problems, always find the unit rate first (cost per item, miles per hour, etc.). This makes the math cleaner and helps you avoid proportion setup errors. Double-check by seeing if your answer makes sense—15 notebooks should cost less than twice what 8 notebooks cost.

Question 11

A recipe calls for milk and cream in a 5:2 ratio. If Jenny uses 3 cups of cream, how much milk should she use to maintain the correct ratio?

7.5 cups of milk (correct answer)
6.0 cups of milk
8.5 cups of milk
5.0 cups of milk
9.0 cups of milk

Explanation: When you encounter ratio problems, you're working with proportional relationships between quantities. The key is setting up a proportion that maintains the same relationship as the original ratio. The recipe calls for milk and cream in a 5:2 ratio, meaning for every 5 parts milk, you need 2 parts cream. Since Jenny uses 3 cups of cream instead of the "2 parts" in the original ratio, you need to find how much milk corresponds to this amount. Set up a proportion:

\frac{\text{milk}}{\text{cream}} = \frac{5}{2} = \frac{x}{3}

Cross multiply:

2x = 5 \times 3 = 15

Solving for x:

x = 7.5

So Jenny needs 7.5 cups of milk, making A correct. Looking at the wrong answers: B (6.0 cups) might tempt you if you mistakenly add 3 to the original milk ratio of 5, but ratios don't work by simple addition. C (8.5 cups) could result from incorrectly multiplying 5 by 1.7 instead of 1.5. D (5.0 cups) represents using the original ratio amount without adjusting for the increased cream—this ignores that Jenny used more cream than the base ratio calls for. Strategy tip: Always set up ratios as fractions and cross multiply to solve. Don't try to "eyeball" the answer or use simple addition/subtraction. The multiplier that changes one part of the ratio must be applied to all parts to maintain the proportional relationship.

Question 12

Two quantities x and y are in direct proportion. When x = 15, y = 45. What is the value of y when x = 28?

84 (correct answer)
75
90
96
72

Explanation: When you see "direct proportion" or "directly proportional," you're dealing with a constant ratio relationship where

y = kx

for some constant

k

. As one quantity increases, the other increases at a steady rate. First, find the constant of proportionality using the given information. When

x = 15

and

y = 45

k = \frac{y}{x} = \frac{45}{15} = 3

So the relationship is

y = 3x

. Now when

x = 28

y = 3 \times 28 = 84

Looking at the wrong answers: Choice B (75) might tempt you if you mistakenly calculated

\frac{45}{15} = 3

, then added this to 25 (thinking 28 - 15 = 13, but making arithmetic errors). Choice C (90) could result from incorrectly thinking the constant is 45 ÷ 15 = 3, then doing 30 × 3 instead of 28 × 3. Choice D (96) might come from setting up an incorrect proportion or confusing direct proportion with some other relationship. The correct answer is A) 84. Study tip: For direct proportion problems, always start by finding the constant ratio

k = \frac{y}{x}

using the given pair of values. Then apply this constant to find unknown values. Remember that in direct proportion, if one variable doubles, the other doubles too—the ratio stays constant throughout.

Question 13

The number of hours students spend on homework is inversely proportional to their efficiency rating. A student with an efficiency rating of 8 spends 3 hours on homework. How many hours would a student with an efficiency rating of 12 spend?

2 hours (correct answer)
1.5 hours
2.5 hours
4.5 hours
3.6 hours

Explanation: When you see "inversely proportional" in a problem, you're dealing with a relationship where one value increases as the other decreases. The key formula is

xy = k

, where

k

is a constant. First, find the constant using the given information. A student with efficiency rating 8 spends 3 hours on homework:

8 \times 3 = 24

. So

k = 24

. Now you can find how many hours a student with efficiency rating 12 would spend. Using the same relationship:

12 \times h = 24

, where

h

is the unknown number of hours. Solving:

h = \frac{24}{12} = 2

hours. Looking at the wrong answers: Choice B (1.5 hours) comes from incorrectly assuming the relationship is

\frac{8}{3} = \frac{12}{h}

, which would be direct proportion rather than inverse. Choice C (2.5 hours) might result from arithmetic errors or confusing the setup. Choice D (4.5 hours) suggests the student thought that higher efficiency means more hours, completely misunderstanding that "inverse" means the opposite relationship. The correct answer is A) 2 hours. Remember this pattern: in inverse proportion problems, always multiply the paired values to find your constant first. Then use that constant with your new given value to find the unknown. Also, do a quick logic check—since efficiency rating increased from 8 to 12, homework time should decrease from 3 hours, which confirms that 2 hours makes sense.

Question 14

A bakery uses flour, sugar, and eggs in the ratio 8:3:2 to make cake batter. If they use 24 pounds of flour, what is the total weight of all ingredients combined?

39.0 pounds total (correct answer)
42.5 pounds total
36.0 pounds total
45.0 pounds total
33.0 pounds total

Explanation: When you encounter ratio problems, you're working with proportional relationships between quantities. The key is understanding that ratios tell you the relative amounts, not the actual amounts, until you're given one specific quantity. The ratio 8:3:2 means for every 8 parts flour, there are 3 parts sugar and 2 parts eggs. Since you know they used 24 pounds of flour, you can find the "unit value" of each part. If 8 parts equal 24 pounds, then each part equals

24 ÷ 8 = 3

pounds. Now you can calculate each ingredient: flour is

8 × 3 = 24

pounds (which matches what you were told), sugar is

3 × 3 = 9

pounds, and eggs are

2 × 3 = 6

pounds. The total weight is

24 + 9 + 6 = 39

pounds, making A correct. Looking at the wrong answers: B (42.5 pounds) might result from incorrectly adding the ratio numbers to the flour amount or making calculation errors. C (36.0 pounds) could come from forgetting to include one ingredient or miscalculating the proportions. D (45.0 pounds) might occur if you mistakenly treated the ratio as actual pounds per ingredient rather than proportional parts. Remember this strategy for ratio problems: first find the value of one "part" by dividing the known quantity by its ratio number, then multiply each ratio component by that unit value. Always double-check that your known quantity matches your calculated result for that ingredient.

Question 15

The ratio of boys to girls in a school club is 4:7. If there are 28 boys in the club, what is the ratio of boys to the total number of club members?

4:11 (correct answer)
28:49
4:7
28:77
1:3

Explanation: When you encounter ratio problems, remember that ratios show relationships between parts, and you can often use them to find other relationships by treating them as fractions. Given that the ratio of boys to girls is 4:7, this means for every 4 boys, there are 7 girls. Since there are actually 28 boys, you can find how many "groups" of the ratio you have:

28 ÷ 4 = 7

groups. This means there are

7 × 7 = 49

girls, making the total membership

28 + 49 = 77

students. The question asks for the ratio of boys to total members, which is

28:77

. However, ratios should be expressed in their simplest form. Since both 28 and 77 are divisible by 7:

28 ÷ 7 = 4

and

77 ÷ 7 = 11

. Therefore, the simplified ratio is 4:11, making A correct. Choice B (28:49) represents the actual number of boys to girls, but the question asks for boys to total members, not boys to girls. Choice C (4:7) is the original boys-to-girls ratio given in the problem, but again, this isn't what's being asked. Choice D (28:77) shows the correct numbers for boys to total members, but it's not simplified to lowest terms. Strategy tip: In ratio problems, always check what relationship the question is actually asking for. The original ratio often appears as a trap answer, and unsimplified ratios are common distractors. Always reduce your final ratio to lowest terms.

Question 16

A rectangle has length and width in the ratio 7:4. If the perimeter is 66 units, what is the area of the rectangle?

252 square units (correct answer)
336 square units
294 square units
168 square units
378 square units

Explanation: When you encounter ratio problems involving perimeter, you need to work with the ratio to find actual dimensions, then calculate what's being asked. Since the length-to-width ratio is 7:4, you can express the dimensions as

7x

and

4x

for some value

x

. The perimeter formula is

P = 2l + 2w

, so:

66 = 2(7x) + 2(4x) = 14x + 8x = 22x

Solving for

x

x = 66 ÷ 22 = 3

This gives you actual dimensions of length =

7(3) = 21

units and width =

4(3) = 12

units. The area is

21 × 12 = 252

square units. Looking at the wrong answers: Choice B (336) likely comes from incorrectly using

x = 4

instead of

x = 3

, giving dimensions of 28 and 16. Choice C (294) might result from calculation errors when working with the perimeter equation. Choice D (168) could come from using half the correct perimeter value or other computational mistakes with the ratios. The correct answer is A. Strategy tip: In ratio problems, always introduce a variable (like

x

) to represent the common factor, then use the given constraint (here, perimeter) to solve for that variable. Double-check by verifying your dimensions maintain the original ratio and produce the stated perimeter before calculating the final answer.

Question 17

A school's enrollment ratio of 6th graders to 7th graders to 8th graders is 5:4:3. If there are 180 students total in these three grades, how many 7th graders are there?

60 seventh graders (correct answer)
75 seventh graders
45 seventh graders
72 seventh graders
90 seventh graders

Explanation: When you encounter ratio problems with totals, you're working with proportional relationships where each part relates to the whole in a specific way. To solve this, first understand what the ratio 5:4:3 means. For every 5 sixth graders, there are 4 seventh graders and 3 eighth graders. Think of this as "ratio units" - you have 5 + 4 + 3 = 12 total ratio units representing 180 actual students. To find the value of each ratio unit, divide the total by the sum of ratio parts:

180 ÷ 12 = 15

students per ratio unit. Since seventh graders represent 4 ratio units, multiply:

4 × 15 = 60

seventh graders. Looking at the wrong answers: Choice B (75) would result if you mistakenly used the sixth graders' ratio (5 units × 15 = 75) instead of the seventh graders' ratio. Choice C (45) occurs if you incorrectly used the eighth graders' ratio (3 units × 15 = 45). Choice D (72) doesn't correspond to any logical error in the ratio calculation and might be included as a distractor. You can verify: 75 sixth graders + 60 seventh graders + 45 eighth graders = 180 total students, and 75:60:45 simplifies to 5:4:3. ✓ Strategy tip: In ratio problems, always add up the ratio parts first to find your total units, then divide the actual total by this sum to find the value of one unit. This systematic approach prevents mix-ups between the different groups.

Question 18

A swimming pool can be filled by two pipes. Pipe A alone can fill the pool in 6 hours, while pipe B alone can fill it in 4 hours. How long will it take to fill the pool if both pipes work together?

2.4 hours exactly (correct answer)
2.0 hours exactly
3.0 hours exactly
5.0 hours exactly
1.5 hours exactly

Explanation: When you encounter work rate problems involving multiple workers (or pipes, machines, etc.) working together, think in terms of rates rather than times. The key insight is that rates add together when working simultaneously. First, convert each pipe's time to a rate. Pipe A fills the pool in 6 hours, so its rate is

\frac{1}{6}

of the pool per hour. Pipe B fills the pool in 4 hours, so its rate is

\frac{1}{4}

of the pool per hour. When both pipes work together, their rates combine:

\frac{1}{6} + \frac{1}{4}

. To add these fractions, find a common denominator of 12:

\frac{2}{12} + \frac{3}{12} = \frac{5}{12}

of the pool per hour. If they fill

\frac{5}{12}

of the pool each hour, the time to fill the entire pool is

\frac{1}{\frac{5}{12}} = \frac{12}{5} = 2.4

hours. Choice A (2.4 hours) is correct. Choice B (2.0 hours) likely comes from incorrectly averaging the times:

\frac{6+4}{2} = 5

, then making an error. Choice C (3.0 hours) might result from using the harmonic mean incorrectly or other computational mistakes. Choice D (5.0 hours) is the arithmetic average of the individual times, which is a common but incorrect approach—you can't simply average work times. Remember: for combined work problems, always convert to rates first, add the rates, then convert back to time. Working together is always faster than the average of individual times.

Question 19

A photocopier can make 180 copies in 6 minutes. At this rate, how many copies can it make in 25 minutes?

750 copies total (correct answer)
900 copies total
720 copies total
600 copies total
840 copies total

Explanation: When you encounter a rate problem like this, you're dealing with proportional relationships. The key is finding the unit rate (how much work gets done per unit of time) and then scaling it up. First, find how many copies the machine makes per minute. Since it makes 180 copies in 6 minutes, divide:

180 ÷ 6 = 30

copies per minute. Now multiply this rate by 25 minutes:

30 × 25 = 750

copies. This confirms answer choice A is correct. Let's examine why the other options are wrong. Choice B (900 copies) represents a common error where students might incorrectly calculate the rate as 36 copies per minute (perhaps by making an arithmetic mistake) and then multiply by 25. Choice C (720 copies) could result from calculating 24 copies per minute instead of 30, possibly from division errors. Choice D (600 copies) might come from using 24 copies per minute or from other calculation mistakes in the setup. You can also solve this using a proportion:

\frac{180 \text{ copies}}{6 \text{ minutes}} = \frac{x \text{ copies}}{25 \text{ minutes}}

. Cross-multiplying gives you

180 × 25 = 6x

, so

x = \frac{4500}{6} = 750

. For rate problems on the SSAT, always start by finding the unit rate first. This makes the math cleaner and helps you avoid calculation errors. Double-check your work by seeing if your answer makes sense—750 copies in 25 minutes means about 4 times as much work as 180 copies in 6 minutes, which feels right since 25 is roughly 4 times 6.

Question 20

On a map, 1.5 inches represents 60 miles. Two cities are 4.5 inches apart on the map. If a car travels at 45 mph, how long will it take to drive between the cities?

4 hours exactly (correct answer)
3 hours exactly
4.5 hours exactly
6 hours exactly
5 hours exactly

Explanation: This problem combines map scale conversion with rate calculations, so you need to work through it step by step to avoid mixing up units. First, find the actual distance between the cities. Set up a proportion using the given scale: if 1.5 inches represents 60 miles, then 4.5 inches represents how many miles?

\frac{1.5 \text{ inches}}{60 \text{ miles}} = \frac{4.5 \text{ inches}}{x \text{ miles}}

Cross multiply:

1.5x = 4.5 \times 60 = 270

x = \frac{270}{1.5} = 180

miles. Now calculate the travel time using the formula: time = distance ÷ speed. With 180 miles at 45 mph:

\text{Time} = \frac{180 \text{ miles}}{45 \text{ mph}} = 4 \text{ hours}

Answer A (4 hours exactly) is correct. Answer B (3 hours) likely comes from incorrectly calculating

\frac{180}{60} = 3

by using 60 (the scale distance) instead of 45 (the actual speed). Answer C (4.5 hours) probably results from using the map distance (4.5 inches) directly as time, skipping the conversion entirely. Answer D (6 hours) might come from dividing 270 by 45 without completing the scale conversion properly. Strategy tip: In multi-step problems like this, always convert everything to real-world units first, then apply the formula. Write down each step clearly to avoid accidentally using map measurements where you need actual distances, or scale numbers where you need given rates.