TACHS Quiz: Percent Ratio And Proportion

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Percent Ratio And ProportionQuestion 1 of 14

What is $12\%$ of $250$ ?

TACHS Quiz

TACHS Quiz: Percent Ratio And Proportion

Practice Percent Ratio And Proportion in TACHS 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 Percent Ratio And Proportion, giving you a quick way to practice the rules, question types, and explanations that matter most for TACHS.

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

What is $12\%$ of $250$ ?

40
25
35
30 (correct answer)

Explanation: When you see "What is X% of Y?", you're being asked to find a part of a whole using percentage multiplication. This is one of the most fundamental percentage calculations you'll encounter. To find

12\%

250

, convert the percentage to a decimal by dividing by 100:

12\% = 0.12

. Then multiply:

0.12 \times 250 = 30

. You can also think of this as

\frac{12}{100} \times 250 = \frac{12 \times 250}{100} = \frac{3000}{100} = 30

. Let's examine why the other answers are incorrect. Choice (A) 40 would result from calculating

16\%

250

instead of

12\%

- this suggests a misreading of the percentage. Choice (B) 25 comes from a common error where students might confuse the setup and calculate

10\%

250

, or possibly mix up which number represents the percentage. Choice (C) 35 would result from calculating

14\%

250

, again indicating a misreading of the given percentage. The correct answer is (D) 30. For percentage problems like this, remember the phrase "of means multiply." When you see "X% of Y," immediately convert the percentage to a decimal and multiply. A quick mental check:

10\%

250

25

, so

12\%

should be slightly more than

25

, making 30 reasonable. Always verify your answer makes sense relative to easy benchmark percentages like

10\%

25\%

Question 2

The ratio of red beads to blue beads in a jar is $18:24$ . What is this ratio expressed in simplest form?

$9:12$
$4:3$
$6:8$
$3:4$ (correct answer)

Explanation: When you encounter ratio problems, your goal is to find the simplest form by dividing both parts of the ratio by their greatest common factor (GCF). To simplify the ratio

18:24

, you need to find the GCF of 18 and 24. Start by listing the factors of each number. The factors of 18 are 1, 2, 3, 6, 9, and 18. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. The greatest factor they share is 6. Divide both parts of the ratio by 6:

18 \div 6 = 3

and

24 \div 6 = 4

. This gives you

3:4

, which is the simplest form since 3 and 4 share no common factors other than 1. Choice A (

9:12

) is incorrect because while you did divide both parts by 2, this isn't fully simplified. You can still divide both 9 and 12 by 3 to get

3:4

. Choice B (

4:3

) reverses the ratio completely—this would represent blue beads to red beads instead of red to blue. Choice C (

6:8

) shows partial simplification by dividing by 3, but you can simplify further by dividing both 6 and 8 by 2. Remember that ratios must maintain their original order and be reduced to the point where the two numbers share no common factors other than 1. Always double-check your GCF by ensuring both resulting numbers can't be divided by the same number again.

Question 3

On a map with a scale of $1\text{ cm}:5\text{ km}$ , two towns are $7.2\text{ cm}$ apart. Approximately how far apart are the towns in kilometers?

12 km
30 km
36 km (correct answer)
1.44 km

Explanation: Map scale problems test your ability to convert between map measurements and real-world distances using proportional reasoning. When you see a scale like

1\text{ cm}:5\text{ km}

, this means every 1 centimeter on the map represents 5 kilometers in reality. To find the actual distance, set up a proportion or use direct multiplication. Since

1\text{ cm} = 5\text{ km}

, you can multiply the map distance by 5:

7.2\text{ cm} \times 5 = 36\text{ km}

. Alternatively, set up the proportion

\frac{1\text{ cm}}{5\text{ km}} = \frac{7.2\text{ cm}}{x\text{ km}}

, then cross-multiply to get

x = 36\text{ km}

. This confirms answer C is correct. Looking at the wrong answers: A) 12 km results from incorrectly dividing 7.2 by something close to 0.6, showing confused scale application. B) 30 km comes from multiplying 7.2 by approximately 4.2, suggesting the student misremembered or misapplied the scale factor. D) 1.44 km results from dividing 7.2 by 5 instead of multiplying—this represents the most common error where students flip the conversion direction. Remember this key principle: when the scale shows map units to real units (like

1\text{ cm}:5\text{ km}

), multiply the map measurement by the scale factor to get the real distance. If you're given real distance and need map distance, then you divide. Always check that your answer makes sense—real distances should be much larger than map distances.

Question 4

In a class, the ratio of boys to girls is $5:7$ . If there are $36$ students in the class, how many are girls?

30
15
18
21 (correct answer)

Explanation: When you encounter ratio problems, you're working with parts of a whole. The ratio

5:7

tells you that for every 5 boys, there are 7 girls, but it doesn't directly tell you the actual numbers. To solve this, first find the total number of ratio parts:

5 + 7 = 12

parts total. Since there are 36 students actual, each ratio part represents

36 ÷ 12 = 3

students. The girls represent 7 parts of the ratio, so there are

7 × 3 = 21

girls in the class. Let's examine why the other answers are incorrect. Choice A (30) would mean there are only 6 boys, giving a ratio of

6:30

1:5

, which doesn't match the given

5:7

ratio. Choice B (15) would mean 21 boys and a ratio of

21:15

7:5

, which reverses the given ratio. Choice C (18) would mean 18 boys and a ratio of

18:18

1:1

, completely different from

5:7

. You can verify the correct answer D (21): if there are 21 girls, there must be

36 - 21 = 15

boys. The ratio

15:21

simplifies to

5:7

when you divide both numbers by 3. Study tip: Always check that your answer maintains the original ratio when simplified. Convert ratios to actual quantities by finding the total parts, then determining what each part represents in the real situation.

Question 5

A recipe calls for sugar and flour in the ratio $2:5$ . If you want to use $3\text{ cups}$ of sugar, how many cups of flour should you use to keep the ratio the same?

6 cups
7.5 cups (correct answer)
5 cups
2.5 cups

Explanation: When you see ratio problems, you're working with proportional relationships where quantities maintain the same relative size. The key is recognizing that if one quantity changes, the other must change proportionally to preserve the ratio. The original ratio is sugar to flour as

2:5

. This means for every 2 parts sugar, you need 5 parts flour. When you want to use 3 cups of sugar instead of 2, you need to find what factor the sugar amount changed by, then apply that same factor to the flour. The sugar increased from 2 to 3 cups, which is a factor of

\frac{3}{2} = 1.5

. To maintain the ratio, multiply the original flour amount by this same factor:

5 \times 1.5 = 7.5

cups of flour. You can verify this using a proportion:

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

. Cross-multiplying gives

2x = 15

, so

x = 7.5

. Choice A (6 cups) incorrectly adds 1 to both parts of the ratio, treating this as addition rather than multiplication. Choice C (5 cups) uses the original flour amount without adjusting for the increased sugar. Choice D (2.5 cups) appears to come from incorrectly setting up the proportion as

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

and solving, which reverses the ratio relationship. For ratio problems, always identify the scale factor by comparing the new amount to the original amount of the known quantity, then apply that same factor to find the unknown quantity. This approach works for any ratio problem you'll encounter.

Question 6

Four identical machines produce $600$ pencils in $3$ hours. At the same rate, how many pencils will six machines produce in $5$ hours?

900
1,200
1,500 (correct answer)
2,250

Explanation: When you encounter problems involving rates and multiple machines or workers, you need to find the rate per unit (per machine, per person, etc.) first, then scale up based on the new conditions. Start by finding how many pencils one machine produces per hour. Four machines make 600 pencils in 3 hours, so four machines make

600 ÷ 3 = 200

pencils per hour. This means one machine produces

200 ÷ 4 = 50

pencils per hour. Now you can answer the question: six machines working for 5 hours will produce

6 × 50 × 5 = 1,500

pencils. The correct answer is C. Let's see where the wrong answers come from. Choice A (900) likely comes from incorrectly calculating

6 × 50 × 3 = 900

- using the original 3 hours instead of the new 5 hours. Choice B (1,200) might result from finding the rate for four machines (200 per hour) and multiplying by 6 hours instead of properly accounting for six machines:

200 × 6 = 1,200

. Choice D (2,250) could come from miscalculating the rate as 75 pencils per machine per hour instead of 50, then computing

6 × 75 × 5 = 2,250

. The key strategy for rate problems is to always break them down to the smallest unit first (rate per machine per hour), then build back up to answer the question. This prevents confusion about which numbers to multiply together and ensures you account for all the changing variables correctly.

Question 7

Two similar triangles have corresponding side lengths in the ratio $3:5$ . If the perimeter of the smaller triangle is $27\text{ cm}$ , what is the perimeter of the larger triangle?

36 cm
30 cm
45 cm (correct answer)
54 cm

Explanation: When you encounter similar triangles, remember that all corresponding measurements scale by the same ratio. This includes side lengths, perimeters, and other linear measurements. Since the triangles are similar with corresponding sides in the ratio

3:5

, their perimeters must also be in this same ratio. Think of it this way: if you multiply each side of the smaller triangle by

\frac{5}{3}

to get the larger triangle, the perimeter (which is just the sum of all sides) gets multiplied by that same factor. To find the larger triangle's perimeter, set up a proportion:

\frac{\text{smaller perimeter}}{\text{larger perimeter}} = \frac{3}{5}

. Substituting the known perimeter:

\frac{27}{\text{larger perimeter}} = \frac{3}{5}

. Cross-multiplying gives us

3 \times \text{larger perimeter} = 27 \times 5 = 135

, so the larger perimeter is

135 \div 3 = 45

cm. This confirms answer C is correct. Looking at the wrong answers: A) 36 cm results from incorrectly adding 9 to the smaller perimeter, perhaps thinking the ratio means "add 3 to get 5." B) 30 cm comes from mistakenly using a

9:10

ratio instead of

3:5

. D) 54 cm appears when students double the smaller perimeter, confusing this with a

1:2

ratio problem. Remember: for similar figures, all linear measurements (perimeters, side lengths, heights) scale by the same ratio. Set up a proportion using this ratio to solve efficiently.

Question 8

A sports drink is $25\%$ pure fruit juice. How many liters of pure fruit juice must be added to $8\text{ L}$ of this drink to create a mixture that is $40\%$ juice?

1.5 L
2 L (correct answer)
3.2 L
4 L

Explanation: This is a mixture problem where you're combining liquids with different concentrations. The key is to track the amount of pure juice before and after adding more. Start by finding how much pure juice is currently in the sports drink. Since the drink is 25% juice:

8 \text{ L} \times 0.25 = 2 \text{ L}

of pure juice. Let

x

= liters of pure juice to add. After adding

x

liters:

Total pure juice = $2 + x$ liters
Total volume = $8 + x$ liters
New concentration = $\frac{2 + x}{8 + x} = 0.40$

Solving this equation:

2 + x = 0.40(8 + x)

2 + x = 3.2 + 0.40x

x - 0.40x = 3.2 - 2

0.60x = 1.2

x = 2

So you need 2 liters of pure juice, making B correct. A) 1.5 L would give you

\frac{3.5}{9.5} \approx 37\%

juice concentration, which is too low. C) 3.2 L represents a common trap - this is the amount of pure juice you'd have if the original 8 L were 40% juice, but it's not what you need to add. D) 4 L would create

\frac{6}{12} = 50\%

juice concentration, which overshoots the target. For mixture problems, always set up an equation where the final amount of the desired substance equals the target percentage times the final total volume. This systematic approach prevents calculation errors.

Question 9

What percent of $64$ is $16$ ?

40%
20%
25% (correct answer)
16%

Explanation: When you see "What percent of X is Y?" you're looking for what percentage Y represents of the whole amount X. This is a fundamental percent relationship that appears frequently on standardized tests. To solve this, you need to set up the equation:

\frac{\text{part}}{\text{whole}} = \frac{\text{percent}}{100}

. Here, 16 is the part and 64 is the whole, so you have

\frac{16}{64} = \frac{x}{100}

. First, simplify the fraction

\frac{16}{64}

. Both numbers are divisible by 16:

\frac{16}{64} = \frac{1}{4} = 0.25

. Converting to a percentage:

0.25 \times 100 = 25\%

. Let's examine why the other answers are incorrect. Choice A (40%) likely comes from incorrectly calculating

\frac{16}{40}

instead of

\frac{16}{64}

, perhaps by misreading the numbers or mixing up the relationship. Choice B (20%) might result from the error

\frac{16}{80}

or from incorrectly simplifying

\frac{16}{64}

. Choice D (16%) represents a common trap where students simply use the numerator as the percentage without doing any calculation. Remember this key pattern: "What percent of A is B?" always means

\frac{B}{A} \times 100

. The number after "of" goes in the denominator, and the number after "is" goes in the numerator. Practice identifying which number represents the whole versus the part, as reversing these is a frequent source of errors on the TACHS.

Question 10

Solve for $x$ if $\dfrac{3}{7}=\dfrac{x}{28}$ .

21
9
12 (correct answer)
84

Explanation: When you see a proportion like this, you're dealing with two equivalent fractions that you can solve using cross multiplication. This is one of the most reliable methods for solving proportions. To solve

\frac{3}{7}=\frac{x}{28}

, cross multiply by multiplying the numerator of each fraction by the denominator of the other fraction:

3 \times 28 = 7 \times x

. This gives you

84 = 7x

. Dividing both sides by 7, you get

x = 12

. You can verify this makes sense:

\frac{3}{7} = \frac{12}{28}

. Since

12 ÷ 4 = 3

and

28 ÷ 4 = 7

, these fractions are indeed equivalent. Looking at the wrong answers: Choice (A) 21 would give you

\frac{21}{28} = \frac{3}{4}

, which is larger than

\frac{3}{7}

. Choice (B) 9 would give you

\frac{9}{28}

, which doesn't reduce to

\frac{3}{7}

since 9 and 28 don't share the right common factors. Choice (D) 84 would give you

\frac{84}{28} = 3

, which is way too large compared to our target fraction of

\frac{3}{7}

. Strategy tip: Always cross multiply when solving proportions - it's faster and more reliable than trying to find equivalent fractions by guessing. After solving, do a quick check by substituting your answer back into the original proportion to make sure both sides are equal.

Question 11

A jacket costs $60 before a markup of $ $15\%$ $. What is the new price after the markup?

$75.00
$66.00
$69.00 (correct answer)
$51.00

Explanation: When you encounter markup problems, you're dealing with percentage increases from an original price. A markup means the price goes up by a certain percentage of the original amount. To find the new price after a

15\%

markup on $60, you need to calculate $

15\%

of $60 and add it to the original price. First, convert the percentage: $

15% = 0.15

. Then multiply:

0.15 \times $60 = $9

. Finally, add this markup to the original price:

$60 + $9 = $69$$. This confirms answer C is correct. Alternatively, you can use the shortcut method: multiply the original price by

(1 + $\text{markup rate}$) = 1.15

. So

\$60 \times 1.15 = \$69

. Looking at the wrong answers: A) $75 represents a $

25\%$$ markup ($$\

60 \times 1.25

), suggesting you might have misread the percentage. B) $66 represents a $$10\%$ $ markup ($ $\$60 \times 1.10$ $), indicating you calculated the wrong percentage. D) $51 represents a $$15\%

discount rather than markup (

\$60 \times 0.85

), showing confusion between markup and markdown. Strategy tip: For markup problems, remember that the final price should always be larger than the original. If you get a smaller number, you've calculated a discount instead. Also, practice the shortcut formula: Original Price × (1 + markup rate) = New Price. This saves time and reduces calculation errors.

Question 12

A store takes $10\%$ off an item and then applies an additional $5\%$ discount to the reduced price. What is the final price of an item that originally cost $200?

$180
$170
$171 (correct answer)
$190

Explanation: When you encounter successive discount problems, remember that each discount applies to the already-reduced price, not the original price. You need to calculate step by step. Let's work through this systematically. Start with the original price of $200. The first discount is $

10\%

off, so you pay

90\%

of the original price:

\

200 \times 0.90 = $180

. Now the second discount of

applies to this reduced price of $180, not the original $200. So you pay $

95\%$$ of $180: $

$180 \times 0.95 = $171$$. Choice A (

180) represents only applying the first

10\%

discount and forgetting about the additional

5\%

discount entirely. Choice B (

170) is what you'd get if you incorrectly subtracted both discounts from the original price:

\$200 - (10\% + 5\%) = \$200 - \$30 = \$170

. This treats the discounts as if they're both applied to the original amount. Choice D (

190) would result from applying only a

5\%

$ discount to the original price, mixing up which discount to apply. Choice C ($171) correctly accounts for both successive discounts. Strategy tip: For successive percent problems, multiply by the decimal equivalents in sequence. Here:

\$200 \times 0.90 \times 0.95 = \$171

. This one-step approach helps you avoid the common trap of adding percentages instead of applying them successively.

Question 13

A phone that originally sells for $480 is advertised at $ $25\%$ $ off. What is the sale price?

$400
$360 (correct answer)
$120
$456

Explanation: When you see "percent off" problems, you're dealing with discount calculations. The key insight is that a discount reduces the original price, so you need to find what remains after subtracting the discount amount. To find the sale price, start by calculating the discount amount:

25\%

\$480 = 0.25 \times \$480 = \$120

. This means the customer saves $120. The sale price is the original price minus the discount: $

\

480 - $120 = $360

. Alternatively, you can think of it this way: if there's a

25%

discount, the customer pays

100% - 25% = 75%

of the original price. So the sale price is

0.75 \times $480 = $360

. Looking at the wrong answers: Choice A (

400) might result from incorrectly calculating $$25\%$$ of

480 as

80 instead of

120, then subtracting that smaller amount. Choice C (

120) is the discount amount itself, not the sale price—this represents what you save, not what you pay. Choice D (

456) could come from mistakenly calculating a much smaller discount, perhaps confusing

25%

with

(since

$480 - $24 = $456

). Remember this pattern: for "percent off" problems, you can either subtract the discount amount from the original price, or multiply the original price by the percentage you still pay. Both methods should give you the same answer, so use whichever feels more natural to you.

Question 14

After a store applies a $20\%$ discount, the sale price of a sweater is $44. What was the original price before the discount?

$35.20
$52.80
$48
$55 (correct answer)

Explanation: When you see discount problems like this, remember that the sale price represents what remains after the discount is taken off the original price. If there's a 20% discount, then the customer pays 80% of the original price. Let's call the original price

x

. After a 20% discount, the customer pays

0.80x = \$44

. To find the original price, divide both sides by 0.80:

x = \frac{\$44}{0.80} = \$55

. You can verify this: 20% of $55 is $11, so $

\

55 - $11 = $44

✓ Now let's see why the other answers are wrong. Choice A (

35.20) is what you'd get if you incorrectly subtracted 20% of

44 from

44 itself: $$\

44 - $8.80 = $35.20$$. This reverses the problem. Choice B (

52.80) results from adding 20% of

44 to

44:

\

44 + $8.80 = $52.80

. This assumes the sale price is the base for calculating the discount, which is incorrect. Choice C ($48) might come from incorrectly thinking that if $44 is after a $4 discount, then the original was $48, but this ignores that the discount must be 20% of the original price. Study tip: In discount problems, always identify what the given amount represents. If it's the sale price, then it equals the original price times (100% - discount rate). Set up your equation accordingly.