Ratios and Proportions
Learning objectives
- Write, simplify and compare ratios.
- Solve proportions by cross-multiplication.
- Share a quantity in a given ratio.
- Use unit rates, direct proportion and map scales.
Ratios
A ratio \(a:b\) compares two quantities. It is the same comparison as the fraction \(\frac ab\). Dividing both parts by their greatest common divisor gives the simplest form.
Move the sliders. The dots show the two quantities; the page gives the simplest form and the equivalent ratio you get by multiplying both parts by the scale factor.
Simplify \(18:24\).
The gcd of 18 and 24 is 6, so \(18:24=3:4\).
Proportions
A proportion states that two ratios are equal: \(\frac ab=\frac cd\).
If \(b,d\neq 0\), then \(\frac ab=\frac cd\) exactly when \(ad=bc\).
Show proof
Multiply both sides of \(\frac ab=\frac cd\) by \(bd\) to get \(ad=bc\). Dividing both sides by \(bd\) reverses the step. ∎
Solve \(\frac35=\frac{x}{20}\).
\(5x=3\cdot 20=60\), so \(x=12\).
Share 56 in the ratio \(3:5\).
There are \(3+5=8\) parts, and each part is \(56\div 8=7\). The shares are \(3\cdot 7=21\) and \(5\cdot 7=35\).
Rates and direct proportion
5 notebooks cost 12.50. What does one cost, and what do 8 cost?
One costs \(12.50\div 5=2.50\), so 8 cost \(8\cdot 2.50=20\).
Two quantities are in direct proportion if their ratio is constant: \(\frac yx=k\), that is \(y=kx\). The number \(k\) is the constant of proportionality.
A car travels 150 km in 2 hours. How far does it go in 5 hours at the same speed?
The speed is \(150\div 2=75\) km/h, so the distance is \(75\cdot 5=375\) km.
A map has scale \(1:50000\). Two towns are 3 cm apart on the map. What is the real distance?
The real distance is \(3\cdot 50000=150000\) cm \(=1.5\) km.
Two quantities are in inverse proportion if their product is constant. If 6 workers need 8 days, then 4 workers need \(\frac{6\cdot 8}{4}=12\) days.
Exercises
Simplify the ratio \(45:60\).
Show solution
The gcd is 15, so \(45:60=3:4\).
Solve \(\frac{7}{x}=\frac{21}{15}\).
Show solution
Cross-multiplying, \(21x=7\cdot 15=105\), so \(x=5\).
Paint is mixed from blue and yellow in the ratio \(2:3\). How much of each is needed for 15 litres?
Show solution
There are 5 parts of 3 litres each: 6 litres of blue and 9 litres of yellow.
Are 4 pens for 6 and 10 pens for 15 in direct proportion?
Show solution
\(\frac64=1.5\) and \(\frac{15}{10}=1.5\), so yes: each pen costs 1.5.
On a map of scale \(1:200000\) two cities are 4.5 cm apart. Find the real distance.
Show solution
\(4.5\cdot 200000=900000\) cm \(=9\) km.
5 machines finish a job in 12 hours. How long would 8 machines take?
Show solution
The work is \(5\cdot 12=60\) machine-hours, so 8 machines need \(60\div 8=7.5\) hours.
Summary
- A ratio compares two quantities and can be simplified like a fraction.
- A proportion is an equality of ratios; solve it by cross-multiplication.
- Direct proportion means a constant ratio; inverse proportion means a constant product.
- Unit rates and map scales are proportions in everyday life.