Finding Equivalent Ratios by Multiplying

Find three ratios equivalent to $2:5$.

Multiply both parts by 2: $2 \times 2 = 4$ and $5 \times 2 = 10$ = $4:10$

Multiply both parts by 3: $2 \times 3 = 6$ and $5 \times 3 = 15$ = $6:15$

Multiply both parts by 4: $2 \times 4 = 8$ and $5 \times 4 = 20$ = $8:20$

Answer: Three equivalent ratios are $4:10$, $6:15$, and $8:20$

Checking if Ratios are Equivalent

Are $6:8$ and $15:20$ equivalent ratios?

Simplify the first ratio: Find GCF of 6 and 8: GCF = 2 $6 \div 2 = 3$ and $8 \div 2 = 4$ = $6:8 = 3:4$

Simplify the second ratio: Find GCF of 15 and 20: GCF = 5 $15 \div 5 = 3$ and $20 \div 5 = 4$ = $15:20 = 3:4$

Compare the simplified ratios: Both simplify to $3:4$ = They are equivalent!

Answer: Yes, $6:8$ and $15:20$ are equivalent ratios because both simplify to $3:4$

Finding a Missing Value

If $3:7 = 12:?$, find the missing value.

Find the multiplier: What times 3 equals 12? $3 \times 4 = 12$, so the multiplier is 4 = Multiplier = 4

Apply the same multiplier to the second part: $7 \times 4 = 28$ = Missing value = 28

Verify the answer: $3:7 = 12:28$ $12 \div 4 = 3$ and $28 \div 4 = 7$ ✓ = Verified!

Answer: The missing value is $28$, so $3:7 = 12:28$

Using a Ratio Table

A recipe uses 2 cups of rice for every 5 cups of water. Complete the ratio table for 4, 6, and 10 cups of rice.

Set up the ratio table: Rice : Water = $2:5$ = Base ratio established

Find water for 4 cups of rice: $2 \times 2 = 4$, so $5 \times 2 = 10$ = 4 cups rice : 10 cups water

Find water for 6 cups of rice: $2 \times 3 = 6$, so $5 \times 3 = 15$ = 6 cups rice : 15 cups water

Find water for 10 cups of rice: $2 \times 5 = 10$, so $5 \times 5 = 25$ = 10 cups rice : 25 cups water

Answer: The ratio table: 2:5, 4:10, 6:15, 10:25

Mistake: Adding the same number to both parts instead of multiplying

Why: Adding changes the ratio's value. If $2:3$ becomes $4:5$ (adding 2), these are NOT equivalent.

Correct: Always multiply or divide both parts by the SAME number. $2:3 = 4:6$ (multiplying by 2).

Mistake: Multiplying only one part of the ratio

Why: To keep ratios equivalent, you must apply the same operation to BOTH parts.

Correct: $2:5 \times 3 = 6:15$, not $6:5$ or $2:15$.

Mistake: Thinking larger numbers mean larger ratios

Why: $4:6$ looks bigger than $2:3$, but they represent the same relationship.

Correct: The size of the numbers does not determine the ratio's value. Always simplify to compare.

Scaling Recipes

Chefs use equivalent ratios to adjust recipes for different numbers of servings.

A recipe for 4 people uses 3 eggs and 2 cups of flour. For 8 people, use 6 eggs and 4 cups of flour (multiply by 2).

Map Scales

Maps use ratios to represent real distances. A scale of 1:50,000 means 1 cm on the map equals 50,000 cm (or 500 m) in reality.

On a 1:100 scale map, 5 cm represents 500 cm (5 meters) in real life.

Mixing Paint

Artists mix colors using specific ratios to get consistent shades.

To make light green, mix blue and yellow in a 1:4 ratio. For 20 ml total, use 4 ml blue and 16 ml yellow.

Q: How are equivalent ratios different from equivalent fractions?

A: They work the same way! A ratio $2:3$ can be written as the fraction $\frac{2}{3}$. Finding equivalent ratios is just like finding equivalent fractions - multiply or divide both parts by the same number.

Q: Can ratios have decimals?

A: Yes, but we usually simplify them to whole numbers. For example, $1.5:3$ is equivalent to $1:2$ (divide both by 1.5) or $3:6$ (multiply both by 2).

Q: What is the simplest form of a ratio?

A: A ratio is in simplest form when both parts share no common factors other than 1. For example, $6:9$ simplifies to $2:3$ by dividing both by their GCF (3).