Learning goals
- Write a ratio in colon, fraction, and “to” form and simplify it
- Set up a proportion and solve for a missing value with cross-multiplication
- Scale quantities up or down while keeping a ratio constant
Ratios compare parts of a whole — or two related amounts
A ratio compares two quantities in the same units (or carefully stated different units). The ratio 3:5 means “3 of the first for every 5 of the second.” You can also write it as 3/5 or “3 to 5.”
Order matters. The ratio of flour to sugar is not the same as sugar to flour unless the amounts are equal. Always name what each part represents.
A proportion says two ratios are equal. If a recipe uses 2 cups flour for 3 cups milk, then 4 cups flour needs 6 cups milk — the ratio stays 2:3. Solving proportions is how you scale maps, recipes, mixtures, and unit rates.
Proportion (cross-multiply)
a/b = c/d ⇒ a·d = b·c
When two ratios are equal, the cross products are equal. If one value is unknown — say a/b = c/x — then a·x = b·c and x = (b·c)/a.
Part from a whole given a ratio
part = whole × (ratio part ÷ sum of ratio parts)
For a ratio m:n that splits a whole W, the first share is W × m/(m+n) and the second is W × n/(m+n).
Example 1 — Simplify a ratio
Given
A class has 12 boys and 18 girls. Write the simplified boys:girls ratio.
Steps
- Write the ratio 12:18.
- Divide both parts by their greatest common factor 6.
- 12 ÷ 6 = 2 and 18 ÷ 6 = 3, so the ratio is 2:3.
Answer
The simplified ratio is 2:3.
Tip: 2:3 means for every 2 boys there are 3 girls — not that there are only 5 students.
Example 2 — Solve a proportion
Given
If 5 notebooks cost $12.50, how much do 8 notebooks cost at the same rate?
Steps
- Set up the proportion: 5/12.50 = 8/x (notebooks over cost).
- Cross-multiply: 5 · x = 12.50 · 8.
- 5x = 100, so x = 20.
Answer
8 notebooks cost $20.00.
Tip: You can also find the unit price first: $12.50 ÷ 5 = $2.50 each, then 8 × $2.50 = $20.00.
Example 3 — Split an amount in a ratio
Given
Share $90.00 in the ratio 2:3 between two people.
Steps
- Total parts = 2 + 3 = 5.
- One part = 90 ÷ 5 = $18.00.
- First person: 2 × 18 = $36.00; second: 3 × 18 = $54.00.
Answer
The shares are $36.00 and $54.00.
Tip: Check: 36 + 54 = 90 and 36:54 simplifies to 2:3.
Reversing the order in a proportion
Common mistake
“5 items cost $12.50, find cost of 8” → set 5/8 = 12.50/x and solve incorrectly for a different meaning.
Better approach
Keep corresponding quantities aligned: items/cost = items/cost, or cost/items = cost/items. Mixing positions breaks the rate.
Check your understanding
1.Simplify 20:35.
Answer: Divide by 5 → 4:7.
2.Solve 3/4 = 9/x.
Answer: 3x = 36 → x = 12.
3.A map scale is 1:50,000. What real distance does 2 cm on the map represent?
Answer: 2 × 50,000 = 100,000 cm = 1 km.
4.Split $120.00 in the ratio 1:5.
Answer: Parts 6; one part $20.00 → $20.00 and $100.00.
Key takeaways
- A ratio compares two quantities; order and units matter.
- Equal ratios form a proportion; solve unknowns with cross-multiplication.
- To split a whole in ratio m:n, divide by (m+n) then multiply by each part.
- Keep corresponding quantities in matching positions when you set up a proportion.