Dividing in a Ratio
"Share £60 between Ana and Ben in the ratio 2 : 3." Not equally, and not £2 and £3. A ratio describes parts. Find out how many parts there are altogether, work out one part, and everything else follows. A bar model makes it almost impossible to get wrong.
National curriculum KS3: use ratio notation, including reduction to simplest form; divide a given quantity into two parts in a given part:part or part:whole ratio; express the division of a quantity into two parts as a ratio; relate ratios to fractions1 · What a ratio says
2 : 3 means "for every 2 of the first, there are 3 of the second". It doesn't say how many there are in total, only how they compare. 2 : 3 could be 2 and 3, or 20 and 30, or 200 and 300. Drag the slider: the picture scales but the ratio stays 2 : 3.
2 · Simplifying a ratio
Like a fraction, a ratio is simplest when its numbers have no common factor. Divide every part by the HCF. 12 : 18 → ÷ 6 → 2 : 3. The groups below show why: 12 teal and 18 orange dots arrange into 6 identical groups, each 2 : 3.
Units first. To simplify 40 cm : 1 m, make the units the same before anything else: 40 : 100 = 2 : 5. A ratio has no units once it's written down. And the order matters: 2 : 3 is not 3 : 2. "Ana to Ben" means Ana's part comes first.
3 · Sharing in a ratio: the bar model
Draw one bar for the whole amount. Split it into as many equal parts as the ratio has in total (2 + 3 = 5). Each part is worth the total ÷ 5. Then Ana gets 2 parts and Ben gets 3. Step through it.
The classic mistake is dividing by the number of people (2) instead of the number of parts (5): "£60 shared 2 : 3, so £30 each, then…". The bar shows five equal parts, not two. Count the parts, always. And check: the shares must add back to the total.
4 · Ratios and fractions
In the ratio 2 : 3, Ana has 2 parts out of 5, which is 2/5 of the whole; Ben has 3/5. A ratio compares part to part; a fraction compares part to whole. Converting between them means knowing whether the denominator should be the total of the parts or just the other part.
5 · When you're given a part, not the total
Harder questions hide the total. "Ana and Ben share money 2 : 3. Ben gets £36. How much in total?" Or: "Ana gets £12 less than Ben. How much does each get?" The bar still works: find out which bit of the bar you've been told, and what one part is worth from that.
6 · Where ratios show up
Recipes and mixing
Squash 1 : 4 with water; mortar 1 : 3 cement to sand; a 2 : 1 : 1 cocktail. Scaling a recipe up or down is multiplying every part by the same number.
Sharing money and prizes
Business partners split profit in the ratio they invested; inheritances and lottery syndicates divide winnings by contribution.
Maps and models
1 : 25 000 is a ratio: map length to real length. The scale factor page is this page with one part always equal to 1.
Screens and photos
Aspect ratio 16 : 9 for a TV, 4 : 3 for an old monitor, 3 : 2 for a camera sensor. A 1920 × 1080 screen is 16 : 9 because both divide by 120.
Gears and bikes
A 48 : 16 chainring-to-sprocket ratio simplifies to 3 : 1: three wheel turns per pedal turn. Gear ratios are simplified ratios.
Computing
Compression ratios (10 : 1), contrast ratios (1000 : 1), and probability odds (3 : 1 against) are all part-to-part comparisons.
Worked examples
Share £84 in the ratio 3 : 4
Simplify 250 g : 1.5 kg
Red and blue paint are mixed 5 : 2. There are 600 ml of red. How much paint in total?
Exam-style: Cara and Dev share sweets 3 : 5. Dev gets 14 more than Cara. How many sweets altogether?
Quick check
What you've learned
A ratio compares parts; it doesn't say how many there are. Simplify by dividing every part by the HCF, after making units match. To share an amount, add the parts, divide the total by that to get one part, then multiply up, and check the shares add back. A part out of the total is a fraction; a part compared with another part is a ratio. When you're told one share or a difference rather than the total, work out one part from what you know and build the rest.