Find the value of one part. Divide the amount you know by the number of parts it stands for, then multiply by the number of parts you want.
Fig. 1: a bar model for sharing $360 in the ratio 2 : 3 : 4. Every block is worth $360 ÷ 9 = $40
Section 01
Ratios in their simplest form.
In the lab, choose Simplify and drag the sliders: the blocks regroup until no common factor is left. Notice that the blocks never change size within one picture, which is why a ratio only makes sense when both parts are in the same units.
Simplify by dividing every part by the highest common factor: \(20 : 30 : 40 = 2 : 3 : 4\).
Same units first: \(1.5\ \text{kg} : 600\ \text{g}\) becomes \(1500 : 600\) before you simplify.
Decimals or fractions: multiply every part by the same number to clear them: \(0.4 : 1.2 = 4 : 12 = 1 : 3\).
The form \(1 : n\): divide both parts by the first one. \(4 : 10 = 1 : 2.5\); here \(n\) need not be a whole number.
Worked example Simplify with mixed units
Write \(1.5\ \text{kg} : 600\ \text{g}\) in its simplest form.
Change to the same unit: \(1.5\ \text{kg} = 1500\ \text{g}\), so the ratio is \(1500 : 600\).
The highest common factor of 1500 and 600 is 300.
\(1500 \div 300 = 5\) and \(600 \div 300 = 2\).
Answer: \(5 : 2\)
Examiner trap
Simplifying \(1.5\ \text{kg} : 600\ \text{g}\) as \(1.5 : 600 = 1 : 400\) is wrong: the parts were in different units. Convert first, then simplify.
Section 02
Dividing in a given ratio.
Add the parts to find how many equal blocks there are. Divide the total by that number to get one part, then build each share. Fig. 1 shows the method as a bar model.
Ana and Ben share some money in the ratio \(5 : 3\). Ana receives \(\$48\) more than Ben. How much money is shared?
The difference between the shares is \(5 - 3 = 2\) parts.
So 2 parts \(= \$48\), and one part \(= \$24\).
The total is \(5 + 3 = 8\) parts: \(8 \times 24 = 192\).
Answer: \(\$192\) (Ana \(\$120\), Ben \(\$72\))
Your turn
Red and green sweets in a jar are in the ratio \(3 : 5\). There are 30 green sweets. How many sweets are there altogether?
Green is 5 parts, so 1 part \(= 30 \div 5 = 6\). The total is \(3 + 5 = 8\) parts: \(8 \times 6 = 48\) sweets.
Examiner trap
To share \(\$360\) in the ratio \(2 : 3 : 4\), do not divide by 2, 3 or 4. Divide by the TOTAL number of parts, 9.
Stage 1 · Learn
Check your understanding
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Write \(12 : 18\) in its simplest form.
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\(\$50\) is shared in the ratio \(3 : 2\). What is the larger share?
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A map has scale \(1 : 50\,000\). What real distance does 4 cm on the map represent?
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3 pens cost \(\$2.10\). What do 7 of the same pens cost?
Section 03
Proportion in context.
Recipes, map scales and best-value questions all use the same idea: find the value for ONE unit (one person, one centimetre, one gram), then scale up. In the lab, try the Recipe and Map scale modes.
Worked example Adapt a recipe
A recipe for 6 people uses 450 g of flour. How much flour is needed for 10 people?
For 1 person: \(450 \div 6 = 75\) g.
For 10 people: \(75 \times 10 = 750\) g.
Answer: 750 g
Worked example Use a map scale
A map has scale \(1 : 25\,000\). Two villages are 6.4 cm apart on the map. Find the real distance in kilometres.
Real distance \(= 6.4 \times 25\,000 = 160\,000\) cm.
\(1\ \text{km} = 1000\ \text{m} = 100\,000\ \text{cm}\), so divide by \(100\,000\).
Answer: 1.6 km
Worked example Best value
Rice is sold in a 500 g bag for \(\$3.40\) and a 750 g bag for \(\$4.95\). Which bag is better value?
Compare the cost of the same amount, say 100 g.
500 g bag: \(\$3.40 \div 5 = \$0.68\) per 100 g.
750 g bag: \(\$4.95 \div 7.5 = \$0.66\) per 100 g.
Answer: The 750 g bag (it is cheaper per 100 g)
Worked example Inverse proportion
6 workers take 10 days to build a wall. How long would 4 workers take, working at the same rate?
Fewer workers take longer, so this is inverse proportion.
Total work \(= 6 \times 10 = 60\) worker-days.
With 4 workers: \(60 \div 4 = 15\) days.
Answer: 15 days
Examiner trap
Map scales hide a units trap: \(1\ \text{km} = 100\,000\ \text{cm}\), not 1000 cm. Write the conversion down before you divide.
Stage 2 · Exam
Exam-style questions
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In a club the ratio of boys to girls is \(4 : 5\). There are 12 more girls than boys. How many members does the club have?
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Amy, Ben and Cal share \(\$480\) in the ratio \(2 : 3 : 5\). How much more does Cal receive than Amy?
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A recipe for 8 people uses 300 g of butter. How much butter is needed for 14 people?
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8 identical pumps empty a tank in 15 hours. How long would 6 of these pumps take?
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