Lesson 34: Proportions and Ratio Word Problems

New Concept

A proportion is a statement that two ratios are equal.

What’s next

Next, you'll learn to solve for missing terms in proportions and apply this skill to organize and solve complex ratio word problems.

Property

A proportion is a statement that two ratios are equal. Equal ratios reduce to the same value or are connected by a constant multiplier. For example,

Examples

• The ratios
  $$\frac{2}{4}$$
  and
  $$\frac{6}{12}$$
  form a proportion because multiplying the first fraction by
  $$\frac{3}{3}$$
  gives you the second.
• A car lot charging 3 dollars for 2 hours and 4 dollars for 3 hours is not proportional because
  $$\frac{3}{2} ≠ \frac{4}{3}$$
  .

Explanation

Think of proportions as two fractions playing dress-up! They look different, but they're the same value in disguise. You can reveal their secret identity by finding the magic number that connects them. This 'constant factor' proves that the relationship between the numbers is consistent and fair.

Property

To find a missing number in a proportion, use a variable and find the multiplier between the terms. In the proportion

Examples

• To solve
  $$\frac{24}{m} = \frac{8}{5}$$
  , notice that $8 \times 3 = 24$. Apply this to the denominator: $5 \times 3 = 15$, so $m=15$.
• To solve
  $$\frac{4}{5} = \frac{16}{y}$$
  , see that $4 \times 4 = 16$. Apply this to the denominator: $5 \times 4 = 20$, so $y=20$.

Explanation

Put on your detective hat! When a number is missing from a proportion, find the relationship between the numbers you do know. If you see $2 \times 3 = 6$, you can apply that same secret rule to the other pair of numbers. It’s a simple trick to solve the mystery every time!

Property

Use a ratio table with two columns, 'Ratio' and 'Actual Count', and two rows for the items being compared. This table helps organize the numbers from a word problem to set up a proportion.

Examples

• The ratio of boys to girls is 3 to 4. If there are 12 girls, how many boys? Set up
  $$\frac{3}{4} = \frac{b}{12}$$
  . Since $4 \times 3 = 12$, then $3 \times 3 = 9$. There are 9 boys.
• A triangle's base-to-height ratio is 3 to 2. If the base is 24 inches, find the height. Set up
  $$\frac{3}{2} = \frac{24}{h}$$
  . Since $3 \times 8 = 24$, then $2 \times 8 = 16$. The height is 16 inches.

Explanation

Word problems can be messy, but a ratio table is your secret weapon for tidiness! It neatly sorts the 'recipe' numbers (the ratio) from the 'real-world' numbers (the actual count). This little grid makes it super easy to build a proportion and solve for the missing piece of the puzzle.