ISEE Ratios and Percentages: Two Ways to Solve One Problem

Prerequisites

This section lists what a student should already be able to do before starting. The lesson assumes comfort with equivalent fractions, with multiplying and dividing decimals, and with the meaning of percent as parts per hundred. A student who can write one fifth as 0.20 and as 20% without hesitation is ready.

The central idea is that a ratio, a fraction, a decimal and a percent are four ways to write the same relationship. The table shows the same single share written four ways.

Share of a whole	Fraction	Decimal	Percent	Part-to-part ratio
1 part out of 4 equal parts	1/4	0.25	25%	1 : 3
1 part out of 5 equal parts	1/5	0.20	20%	1 : 4
3 parts out of 10 equal parts	3/10	0.30	30%	3 : 7
3 parts out of 4 equal parts	3/4	0.75	75%	3 : 1

The last column is where many errors begin. A ratio such as 3 : 7 compares one group with another group, so the whole is 3 + 7 = 10 parts, and the first group is 3/10 of the whole, not 3/7.

A worked example with two solution paths

This section solves one situation twice, to show that the ratio method and the percentage method are interchangeable. A painter mixes blue and white paint in the ratio 3 : 7 and needs 20 liters of mixture in total. The question asks how many liters of blue paint are needed.

Path 1: the ratio method.

1. Add the parts to find the whole: 3 + 7 = 10 equal parts.
2. Divide the total by the number of parts: 20 ÷ 10 = 2 liters per part.
3. Multiply by the blue share: 3 × 2 = 6 liters.

Path 2: the percentage method.

1. Convert the ratio to a share of the whole: blue is 3/10, which is 0.30 or 30%.
2. Multiply the total by that share: 0.30 × 20 = 6 liters.

Both paths give 6 liters, and either can verify the other. A third route sets up a proportion, 3/10 = b/20, and cross-multiplies to get 10b = 60, so b = 6. Students should choose the path that makes the arithmetic simplest for the numbers in front of them.

The same flexibility helps when a percent is given and a ratio is easier to use. Suppose a jacket is marked down 25% and the sale price is $60. The question asks for the original price.

• Percentage path: the sale price is 100% − 25% = 75% of the original, so 0.75 × original = 60 and the original is 60 ÷ 0.75 = $80.
• Ratio path: 75% is 3/4, so the sale price and original price are in the ratio 3 : 4. Three parts equal $60, so each part is $20 and the original price of four parts is $80.

A quick check closes the loop: 25% of $80 is $20, and $80 − $20 = $60.

Try the example

This section points to the interactive item. Work the original question below on paper first, choose an answer, and then read the explanation for every option; the item is an original practice item written by the ISEEPrep teaching team and is not a question from the actual ISEE.

Why each answer works or does not

This section explains how the four choices relate to the two methods. The item has two steps: find the number of seventh graders from the 3 : 5 ratio, then find 20% of that group.

Using the ratio method, 3 + 5 = 8 parts, 80 ÷ 8 = 10 students per part, and the seventh graders number 5 × 10 = 50. One fifth of 50 is 10. Using the percentage method, 5/8 is 62.5%, 0.625 × 80 = 50, and 0.20 × 50 = 10. Both methods agree, which supports the answer.

Each wrong choice is the result of a specific, predictable slip, and recognizing the slip is more useful than memorizing the item.

• Stopping one step early. Choice (A), 50, is a correct intermediate result. Underlining the final question, “how many are new,” before starting prevents the student from answering the question that was not asked.
• Treating a part as the whole. Choice (B), 16, applies 20% to all 80 students. The percent belongs to the seventh graders only, so the base of the percent is 50.
• Confusing percent with decimal. Choice (D), 1, comes from writing 20% as 0.02. A student can test the form by remembering that 100% is 1.00, so 20% must be 0.20.

Common mistakes

This section collects the errors most often seen in ratio and percent problems at the Middle Level.

1. Using the wrong total. In a 3 : 5 ratio, the whole is 8 parts, not 5. Writing the total as the sum of the parts is the first step of nearly every ratio problem.
2. Applying a percent to the wrong base. Before multiplying, a student should name the group the percent refers to. “20% of the seventh graders” and “20% of the program” are different quantities.
3. Adding a percent back incorrectly. After a 25% discount, the original price is not the sale price plus 25% of the sale price. The percent is taken from the original, so the sale price is 75% of it.
4. Misplacing the decimal point. Converting 5% to 0.5 instead of 0.05 changes the answer by a factor of ten. Estimating first, for example that 5% of 80 is a small number, exposes the error.
5. Skipping the reasonableness check. A result larger than the whole group, or a fraction of a person, signals a setup error. A two-second estimate often catches it.

Your next step

This section suggests where to go after the lesson. Students can review the full range of topics tested in Mathematics Achievement, or compare the reasoning demands of that section with Quantitative Reasoning and the overall structure of the Middle Level. A student who wants to see how this skill performs under timed conditions can take the free trial, which includes a full-length practice test, and the results can show whether percent and ratio items are a strength or a repeated source of lost points.