Ratios, Proportions, and Percent Foundations

1.3: Ratios, Proportions, and Percent Foundations. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 3 of 53 parts Practice Flashcards

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Ratio parts

Objective: Convert a part-to-part ratio to a part-to-whole fraction.

Prerequisite skills: fraction equivalence and decimal arithmetic. A ratio a:b compares quantities in a stated order. If the two categories exhaust the whole, there are a + b total parts, so the first category is a/(a + b) of the total. Common error: using the second part alone as the whole. Restrictions and interpretation: Keep the ratio order and identify the whole used as the percent base.

Formulas

first share = a/(a + b)

Worked examples

Try itRed and blue tokens are in the ratio 23:3. What fraction of all tokens is red?Show answer

There are 23+3 = 26 total parts. Red accounts for 23 of them, so the fraction is 23/26. Therefore the requested answer is 23/26.

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Ratio parts

Objective: Convert a part-to-part ratio to a part-to-whole fraction.

Formulas

first share = a/(a + b)

Worked examples

Try itRed and blue tokens are in the ratio 23:3. What fraction of all tokens is red?Show answer

There are 23+3 = 26 total parts. Red accounts for 23 of them, so the fraction is 23/26. Therefore the requested answer is 23/26.

Solving a proportion

Objective: Find an unknown value in equal ratios.

A proportion states that two ratios are equal. Cross-products are equal when denominators are nonzero, or you can scale both parts by the same factor. Proportional relationships have a constant quotient and pass through the origin. Common error: adding the same number to numerator and denominator. Restrictions and interpretation: Keep the ratio order and identify the whole used as the percent base.

Formulas

$a/b = c/d ⇒ ad = bc$

Worked examples

Try itIf x/6 = 25/9, what is x?Show answer

Multiply both sides by 6: x = 6(25)/9 = 50/3 = 50/3. Therefore the requested answer is 50/3.

Percent as a multiplier

Objective: Find a percentage of a whole.

Percent means per hundred. Convert p% to p/100, then multiply by the whole. To find a missing whole, divide the known part by the percent expressed as a decimal. Common error: using p rather than p/100 as the multiplier. Restrictions and interpretation: Keep the ratio order and identify the whole used as the percent base. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

$part = (p/100) × whole$

Worked examples

Try itA club has 540 members. 125% volunteer. How many members volunteer?Show answer

Convert 125% to 1.25. Multiply by 540: (125/100)(540) = 675 members. Therefore the requested answer is 675.

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