Ratio and proportion.
Find one part, scale the relationship and keep quantities in proportion.
Simplify ratios · divide quantities · adapt recipes · use scales · compare value
A ratio compares quantities
Imagine mixing juice with 2 cups of concentrate and 3 cups of water. The ratio 2 : 3 describes the relative amounts, not the total amount of drink. Doubling both amounts keeps the same taste because the comparison stays unchanged.
A ratio therefore behaves like a recipe in parts. To simplify it, divide every part by the same common factor. If the quantities use different units, convert them first; 1 metre compared with 50 centimetres is not the ratio 1 : 50.
A ratio records relative amounts in a stated order. 2 : 3 means two parts of the first quantity for every three parts of the second.
Simplify 20 : 30 : 40 and 1.5 m : 75 cm.
Divide every part of 20 : 30 : 40 by their HCF, 10: 2 : 3 : 4.
Use matching units before simplifying: 1.5 m : 75 cm = 150 cm : 75 cm = 2 : 1.
For fractions, clear denominators first: 12 : 34 = 2 : 3 (multiply both parts by 4). Equivalent ratios keep the same relative amounts: 2 : 3 = 4 : 6 = 10 : 15.
Pause and explainIn a ratio 2 : 3, how many equal parts are there altogether?
Five. The whole mixture contains two parts of the first ingredient and three parts of the second.
Share the total using equal parts
Sharing $40 in the ratio 3 : 5 does not mean giving one person $3 and the other $5. It means splitting all $40 into eight equal parts, then distributing three parts and five parts.
Find the value of one part by dividing the total by the sum of the ratio numbers. Multiply that value by each person’s number of parts. Adding the shares back together checks the total; comparing them checks that the requested ratio was preserved.
Total = 150. Five equal parts, so one part = 150 ÷ 5 = 30.
Share $360 in the ratio 2 : 3 : 4.
- Add the parts: 2 + 3 + 4 = 9.
- Find one part: $360 ÷ 9 = $40.
- Multiply: shares are $80, $120 and $160.
- Check: 80 + 120 + 160 = 360 and 80 : 120 : 160 simplifies to 2 : 3 : 4.
Red : blue beads = 2 : 5. There are 35 blue beads. Find red and total.
The known 35 corresponds to 5 parts, so one part is 7.
Red = 2 × 7 = 14. Total = 7 parts = 49 beads.
Pause and explainWhat is one part when $40 is shared in the ratio 3 : 5?
$5, because there are 8 parts and 40 ÷ 8 = 5. The shares are $15 and $25.
Scale recipes and read maps
A recipe can be scaled because every ingredient follows the same multiplier. A map uses the same idea: a fixed drawing length stands for a corresponding real length.
First identify the two matching quantities. For a recipe, compare the new serving count with the old one. For a map scale, check whether both lengths are in the same unit. Once the common multiplier is known, apply it to each relevant quantity, not just the easiest one.
A recipe for 4 uses 300 g flour. Adapt it for 10.
Scale factor = 104 = 2.5.
Multiply every ingredient by the same factor.
A map scale is 1 : 50 000. What does 3.6 cm represent?
1 cm on the map = 50 000 cm in reality = 0.5 km.
In reverse, 4 km = 400 000 cm, so the map length is 400 000 ÷ 50 000 = 8 cm.
Pause and explainIf a recipe for 4 people is adapted for 6, what multiplier applies?
6 ÷ 4 = 1.5. Every ingredient amount must be multiplied by 1.5.
Compare value using one common unit
A larger pack can cost more overall and still provide better value. To compare offers fairly, compare equal amounts: for example, price per kilogram or grams obtained per dollar.
Choose one common unit and calculate the corresponding rate for both products. If using cost per unit, the smaller result is better value. If using quantity per dollar, the larger result is better. Say which rate you used so the comparison is clear.
Rice: 750 g for $2.40 or 1.2 kg for $3.60. Which is better value?
Convert 750 g to 0.75 kg.
Pack A
Pack B
Pack B is cheaper per kilogram. A lower cost per unit means better monetary value when the products are comparable.
Pause and explainWhy is comparing a 500 g price directly with a 1 kg price misleading?
They buy different amounts. Convert to a shared quantity, such as the price per kilogram, before choosing.
Your turn. Find one part first.
Read whether the question gives a total, one share, or a scale relationship. Add ratio parts only when you need the total number of equal shares.
Check units before simplifying, show the value of one part when sharing, and identify your comparison rate for best-value questions. Finish with a sentence explaining what each calculated amount represents.
Simplify 0.6 : 1.5 : 2.1.
Reveal worked answer
Share 420 in the ratio 3 : 4 : 5.
Reveal worked answer
A recipe uses 180 ml milk for 6 servings. Find the milk needed for 15 servings.
Reveal worked answer
On a 1 : 25 000 map, a path is 7.2 cm long. Find its actual length in km.
Reveal worked answer
Which is better value: 400 g for $1.80 or 900 g for $3.78?