TL;DR:
A ratio compares two quantities. A proportion says two ratios are equal.
On a 1:50,000 map, 1 cm stands for 50,000 cm, which is 0.5 km.
Write the units next to each number and cancel them. They tell you if the setup is right.
A hiking map says its scale is 1:50,000. Two trailheads are 3.2 cm apart on the map. How far is the walk in a straight line? A recipe makes 12 cookies from 3 cups of flour. How much flour for 20? Both are proportions, and one habit of setting them up solves them.
Ratios, rates and proportions
A ratio compares two amounts, such as 1 to 50,000, or 3 cups to 12 cookies. A rate is a ratio of different units, such as 4 cookies per cup. A proportion is a statement that two ratios are equal, such as 3/12 = 5/20. If one quantity is multiplied by a number, the other is multiplied by the same number.
Worked example: a 1:50,000 map scale
The scale 1:50,000 means any length on the map is 50,000 times smaller than in real life. Both numbers are in the same unit, so 1 cm on the map is 50,000 cm on the ground.
Set the map distance: 3.2 cm.
Multiply by the scale:
3.2 × 50,000 = 160,000cm on the ground.Convert to metres:
160,000 ÷ 100 = 1,600m.Convert to kilometres:
1,600 ÷ 1,000 = 1.6km.
Check the other way. 1 cm on this map is 50,000 cm = 500 m = 0.5 km. Then 3.2 cm is 3.2 × 0.5 = 1.6 km. Both routes agree.
Now reverse it. A trail is 4.5 km long in real life. In centimetres that is 4.5 × 1000 × 100 = 450,000. Divide by 50,000: 450,000 ÷ 50,000 = 9 cm on the map. Real distances are bigger than map distances, so the real number should always be the larger one. Here 450,000 is bigger than 9, which is as it should be.
Recipes: scale everything by one factor
The cookie recipe: 3 cups of flour for 12 cookies. For 20 cookies, set up the proportion with matching units on each side: 3 cups / 12 cookies = x cups / 20 cookies.
Cross multiply: 3 × 20 = 12 × x, so 60 = 12x and x = 5. You need 5 cups. Check with the unit rate: 12 cookies from 3 cups is 12 ÷ 3 = 4 cookies per cup, so 20 cookies need 20 ÷ 4 = 5 cups. Same answer.
Why does cross multiplication work? If a/b = c/d, multiply both sides by b × d: the left becomes a × d and the right c × b. It is the normal equation rule of doing the same to both sides, in a shortcut.
Unit rates: the fair comparison
To compare two options, convert both to the same unit rate. Say a 500 g pack costs $3.00 and a 750 g pack costs $4.20 (made-up prices). Per gram: 3.00 / 500 = 0.006 and 4.20 / 750 = 0.0056. The larger pack is cheaper per gram. Check by scaling the small pack to 750 g: 3.00 × 1.5 = 4.50, which is more than $4.20. Consistent.
Try one yourself: mixing paint
A paint mix uses 2 parts blue to 3 parts white, and you need 15 litres in total. There are 2 + 3 = 5 parts, so one part is 15 ÷ 5 = 3 litres. Blue is 2 × 3 = 6 litres and white is 3 × 3 = 9 litres. Check: 6 + 9 = 15, and 6 : 9 simplifies to 2 : 3.
When more of one means less of the other
Not every proportion goes the same way. If 6 workers finish a job in 8 days (a made-up job), and all work at the same pace, the total work is 6 × 8 = 48 worker-days. With 4 workers it takes 48 ÷ 4 = 12 days. More workers means fewer days, so the product stays fixed rather than the ratio. This is called inverse proportion. Check the direction before choosing a method: if one quantity goes up and the other should go down, multiplying by the same factor is wrong.
How to check your answer
Units cancel. In
3.2 cm × 50,000the answer is in the ground unit (cm), and you then convert.Direction. If the map is smaller than the world, a map-to-real answer should grow.
Reverse it. Convert your answer back and see if you recover the start.
Two methods. Try a unit rate as well as cross multiplication.
Common mistakes
Mixing units in the scale. 1 cm to 50,000 m is a very different thing from 1 cm to 50,000 cm.
Putting numbers on the wrong sides, such as
cups / cookies = cookies / cups.Dividing when the problem needs multiplication. The direction check catches this.
Forgetting to convert to a friendly unit at the end, leaving 160,000 cm instead of 1.6 km.
Assuming every relationship is proportional. A fixed fee plus a rate, as in a phone plan, is not.
Where you will use it next
Scale factors return in shapes of the same form, as in similar triangles: measuring a tree with its shadow. When a quantity changes by a fixed multiple, the guide to rises and falls shows the same idea written as a multiplier.
Frequently asked questions
Is a ratio the same as a fraction?
They are written alike, but a ratio compares two amounts, while a fraction usually describes a part of one whole. The arithmetic is the same.
How do I know if a relationship is proportional?
Check that the ratio of the two quantities is always the same. If it changes, such as with a fixed starting fee, it is not proportional.
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