Math Foundations Unit 5: How-To Guide

Part 1: Ratios, Rates, & Proportions

How to Write a Ratio

  1. Identify the two numbers or items you need to compare.
  2. Write the ratio in the order the question asks for (e.g., "cats to dogs" is different from "dogs to cats").
  3. Write the ratio using a colon (3:4), the word "to" (3 to 4), or as a fraction (\(\frac{3}{4}\)).
  4. Simplify the ratio just like you would simplify a fraction.

Worked Example:

A classroom has 12 boys and 18 girls. What is the ratio of boys to girls?

Step 1 & 2: We are comparing boys to girls, so the order is 12 to 18.

Step 3 (Write): \( \frac{12}{18} \)

Step 4 (Simplify): Both numbers can be divided by 6. \( \frac{12 \div 6}{18 \div 6} = \frac{2}{3} \). The ratio is 2 to 3.

How to Find a Unit Rate

  1. Write the comparison as a rate (a fraction with units).
  2. Divide the top number by the bottom number.
  3. Write the answer with the word "per" (e.g., miles per hour, dollars per pound).

Worked Example:

You drive 180 miles in 3 hours. What is your speed in miles per hour?

Step 1 (Write Rate): \( \frac{180 \text{ miles}}{3 \text{ hours}} \)

Step 2 (Divide): \( 180 \div 3 = 60 \)

Step 3 (Add Units): The unit rate is 60 miles per hour (or 60 mph).

How to Solve a Proportion

  1. Set up the two equal fractions (ratios). Make sure the units match on the top and bottom.
  2. Cross-multiply by multiplying the numbers that are diagonal from each other.
  3. Set the two products equal to each other.
  4. Solve the equation for the missing variable (x).

Worked Example:

If 4 pencils cost $1.00, how much do 10 pencils cost? Let 'x' be the unknown cost.

Step 1 (Set up): \( \frac{4 \text{ pencils}}{\$1.00} = \frac{10 \text{ pencils}}{x} \)

Step 2 & 3 (Cross-Multiply): \( 4 \cdot x = 1.00 \cdot 10 \implies 4x = 10 \)

Step 4 (Solve): \( x = \frac{10}{4} \implies x = 2.5 \). So, 10 pencils cost $2.50.

Part 2: Percent Applications

How to Calculate a Discount

  1. Convert the discount percent to a decimal (divide by 100).
  2. Multiply the decimal by the original price. This gives you the discount amount.
  3. Subtract the discount amount from the original price to get the final sale price.

Worked Example:

A $40 sweater is on sale for 20% off. What is the sale price?

Step 1 (Percent to Decimal): \( 20\% \rightarrow 0.20 \)

Step 2 (Find Discount): \( 0.20 \times \$40 = \$8 \). The discount is $8.

Step 3 (Subtract): \( \$40 - \$8 = \$32 \). The sale price is $32.

How to Calculate Tax or Tip

  1. Convert the tax or tip percent to a decimal (divide by 100).
  2. Multiply the decimal by the original cost. This gives you the tax/tip amount.
  3. Add the tax/tip amount to the original cost to get the final total.

Worked Example:

Your meal at a restaurant costs $50. You want to leave an 18% tip. What is the total cost?

Step 1 (Percent to Decimal): \( 18\% \rightarrow 0.18 \)

Step 2 (Find Tip): \( 0.18 \times \$50 = \$9 \). The tip is $9.

Step 3 (Add): \( \$50 + \$9 = \$59 \). The total cost is $59.