Chapter 6 Study Guide: Ratio, Proportion, and Percent

Ratio, Proportion, and Percent

Section 6.1 — Ratio and Proportion

Ratios and proportions are fundamental concepts used to compare quantities and solve problems involving relationships between numbers. Understanding how to set up and solve proportions is essential for many real-world applications.

• Ratio: A ratio is a comparison of two quantities, expressed as a quotient. The order of the numbers matters: the first quantity is the numerator, the second is the denominator.

• Ways to Write Ratios:

  • a to b

  • a : b

  • a/b

• Reducing Ratios: Always simplify ratios to their lowest terms.

• Proportion: A proportion is an equation stating that two ratios (or rates) are equal.

Proportion Equation:

$a/b = c/d$

This is read as "a is to b as c is to d."

• Solving Proportions — Cross Products Rule: To solve for an unknown in a proportion, set the cross products equal and solve the resulting equation.

$\text{If } \frac{a}{b} = \frac{c}{d}, \text{ then } a \cdot d = b \cdot c$

• Multiply the numerator of one ratio by the denominator of the other; set the two products equal.

Example: Solve for n in $\frac{51}{34} = \frac{3}{n}$

• Set cross products: $51 \cdot n = 34 \cdot 3$

• $51n = 102$

• $n = \frac{102}{51} = 2$

• Applications of Proportions:

  • Set up proportions so corresponding quantities occupy the same position in both ratios.

  • Keep units consistent and label each part.

  • Solve using cross products and answer with correct units.

Section 6.3 — Solving Percent Problems with Equations

Percent problems can be solved by translating sentences into equations using key words. Recognizing these words helps set up the correct mathematical relationship.

• Key Translation Words:

  • "of" → multiplication ($\times$)

  • "is" → equals ($=$)

  • "what" → unknown number (n)

• The Percent Equation:

$\text{percent} \times \text{base} = \text{amount}$

• amount: The part

• base: The whole (the number the percent is "of")

• percent: Must be written as a decimal or fraction before multiplying

Translating & Solving Patterns:

• "15 is what percent of 20?" → $15 = n \times 20$

• "2.4 is 25% of what number?" → $2.4 = 0.25 \times n$

• "What number is 50% of 0.6?" → $n = 0.5 \times 0.6$

• "What number is 45% of 80?" → $n = 0.45 \times 80 = 36$

• "65% of 90 is what number?" → $n = 0.65 \times 90 = 58.5$

• "What percent of 35 is 21?" → $n \times 35 = 21 \Rightarrow n = \frac{21}{35} = 0.6 = 60\%$

Steps to Solve Any Percent Equation Problem:

1. Identify the base (whole), the amount (part), and the unknown.

2. Translate the sentence into an equation using: of = $\times$, is = $=$, what = n.

3. Convert any percent to a decimal (move the decimal point two places left).

4. Solve the equation for n.

5. If solving for a percent, convert the decimal answer back to a percent (move the decimal point two places right and add %).

Section 6.5 — Applications of Percent

Percent applications involve identifying the three parts of a percent problem and solving using either the percent equation or the percent proportion. These methods are used in real-world contexts such as wage calculations, attendance, and changes in quantities.

• Parts of a Percent Problem:

  • Base: The whole amount (the number the percent is "of")

  • Amount: The part (the result of taking the percent of the base)

  • Percent: The rate, written with a % sign or as a decimal

Method 1: The Percent Equation

$\text{percent} \times \text{base} = \text{amount}$

• Translate the sentence directly (of = $\times$, is = $=$, what = n), write the percent as a decimal, then multiply.

• Example: 18% of Frank's $3680 wages are withheld.$ 0.18 \times 3680 = n \Rightarrow n = 662.40 $

Method 2: The Percent Proportion

$\frac{\text{amount}}{\text{base}} = \frac{\text{percent}}{100}$

• Set the amount over the base equal to the percent over 100, then solve using cross products.

• Example: 31 absent students is 4% of the total. $\frac{31}{b} = \frac{4}{100} \Rightarrow 31 \times 100 = 4b \Rightarrow b = 775$ students

Percent of Increase

Percent of increase measures how much a quantity has grown relative to its original value.

$\text{percent of increase} = \frac{\text{amount of increase}}{\text{original amount}}$

• Find the amount of increase: new amount − original amount

• Divide by the original amount and write the quotient as a percent

• Example: Drink size grows from 13.1 oz to 19.9 oz. Increase = $19.9 - 13.1 = 6.8$ Percent increase = $\frac{6.8}{13.1} \approx 0.5191 \approx 51.9\%$

Percent of Decrease

Percent of decrease measures how much a quantity has reduced relative to its original value.

$\text{percent of decrease} = \frac{\text{amount of decrease}}{\text{original amount}}$

• Find the amount of decrease: original amount − new amount

• Divide by the original amount and write the quotient as a percent

• Example: Calories decrease from 3250 to 2100. Decrease = $3250 - 2100 = 1150$ Percent decrease = $\frac{1150}{3250} \approx 0.3538 \approx 35.4\%$