A ratio, a fraction and a linear function are three ways of describing the same multiplicative link. Being able to translate between them turns hard-looking questions into easy ones.
| Language | Written as | Says |
|---|---|---|
| Ratio | $y : x = 3 : 4$ | For every $4$ of $x$ there are $3$ of $y$ |
| Fraction | $y = \dfrac{3}{4}x$ | $y$ is three quarters of $x$ |
| Linear function | $y = 0.75x$ | A straight-line graph through the origin, gradient $0.75$ |
- Write the ratio in the order given, e.g. $y : x = a : b$.
- Rewrite as a fraction equation: $\dfrac{y}{x} = \dfrac{a}{b}$.
- Multiply both sides by $x$: $y = \dfrac{a}{b}x$.
- Convert $\dfrac{a}{b}$ to a decimal if the question prefers it.
In a bakery the ratio of loaves $L$ to rolls $R$ baked each day is $2 : 5$. Write $R$ in terms of $L$.
So there are always $2.5$ times as many rolls as loaves. If $L = 40$, then $R = 100$.
$a : b : c = 3 : 5 : 8$. Write $b$ and $c$ in terms of $a$.
The "let one part be $t$" trick is extremely useful — remember it.
Because $y = \dfrac{a}{b}x$ has no constant term, its graph is a straight line through the origin. The gradient is the ratio.
A straight line through the origin passes through the point $(12,\ 20)$. Write the ratio $y : x$ in its simplest form and state the equation.
Equation: $y = \dfrac{5}{3}x$
The other essential translation is between a ratio and the fractions of the total.
$A = \dfrac{a}{a+b}$ of the total and $B = \dfrac{b}{a+b}$ of the total.
(a) The ratio of cats to dogs is $4 : 7$. What fraction of the animals are dogs?
(b) $\dfrac{3}{10}$ of a class are left-handed. What is the ratio left : right?
When the ratio changes, algebra is the cleanest route. The standard setup is: let one part be $x$.
The ratio of Ann's money to Bob's is $5 : 2$. Ann gives Bob £$18$, and now the ratio is $4 : 3$. How much did Ann have at the start?
Check: Ann £$90$, Bob £$36$ ($5:2$ ✓). After: £$72$ and £$54$, and $72 : 54 = 4 : 3$ ✓
$x : y = 3 : 5$ and $x + y = 96$. Find $x$ and $y$ using algebra.
Check: $36 : 60 = 3 : 5$ ✓ (This matches the "add-divide-multiply" method: $96 \div 8 = 12$, so $3\times12=36$, $5\times12=60$.)
Coordinate geometry uses ratios to find points along a line segment.
$A$ is $(2,\ 3)$ and $B$ is $(14,\ 19)$. Find the point $P$ on $AB$ such that $AP : PB = 1 : 3$.
Ratio → equation
$y : x = a : b \Rightarrow y = \dfrac{a}{b}x$.
Gradient
For $y = \dfrac{a}{b}x$ the gradient is $\dfrac{a}{b}$ — the ratio itself.
Through the origin
Proportional relationships always pass through $(0,0)$.
Ratio → fraction of total
$a : b$ gives $\dfrac{a}{a+b}$ and $\dfrac{b}{a+b}$.
Fraction → ratio
$\dfrac{3}{10}$ are $A$ $\Rightarrow A : B = 3 : 7$, not $3 : 10$.
Let one part be $x$
The universal setup for changing-ratio problems.
Cross-multiply
$\dfrac{p}{q} = \dfrac{r}{s} \Rightarrow ps = qr$.
Dividing a line
$AP : PB = m : n$ means $P$ is $\dfrac{m}{m+n}$ of the way from $A$ to $B$.
$y : x = 7 : 4$. Write $y$ in terms of $x$, and find $y$ when $x = 20$.
▶ Show solution
$\dfrac{y}{x} = \dfrac{7}{4}$, so $y = \dfrac{7}{4}x = 1.75x$.
When $x = 20$: $y = 1.75 \times 20 = 35$.
A straight line through the origin passes through $(15,\ 9)$. Write its equation and the ratio $y : x$ in simplest form.
▶ Show solution
Gradient $= \dfrac{9}{15} = \dfrac{3}{5} = 0.6$.
Equation: $y = 0.6x$ (or $y = \tfrac{3}{5}x$).
Ratio $y : x = 9 : 15 = 3 : 5$.
$\dfrac{4}{9}$ of the members of a club are female. Write the ratio of female to male members.
▶ Show solution
Female $= 4$ parts out of $9$, so male $= 9 - 4 = 5$ parts.
Ratio female : male $= 4 : 5$
$p : q : r = 2 : 7 : 4$. Write $q$ and $r$ in terms of $p$.
▶ Show solution
Let one part be $t$: $p = 2t$, $q = 7t$, $r = 4t$.
$t = \dfrac{p}{2}$
$q = 7 \times \dfrac{p}{2} = \dfrac{7p}{2} = 3.5p$
$r = 4 \times \dfrac{p}{2} = 2p$
Is the relationship $y = 3x + 2$ a proportional one? Explain your answer using the table below.
| $x$ | $1$ | $2$ | $4$ |
|---|---|---|---|
| $y$ | $5$ | $8$ | $14$ |
▶ Show solution
$\dfrac{y}{x}$: $5 \div 1 = 5$, $8 \div 2 = 4$, $14 \div 4 = 3.5$
The ratio is not constant, so no, it is not proportional.
The graph is a straight line but it crosses the $y$-axis at $(0, 2)$, not at the origin.
$a : b = 4 : 9$ and $a + b = 78$. Find $a$ and $b$ using algebra.
▶ Show solution
Let one part be $x$: $a = 4x$, $b = 9x$.
$4x + 9x = 78 \Rightarrow 13x = 78 \Rightarrow x = 6$
$a = 24$, $b = 54$. Check: $24 + 54 = 78$ ✓
The ratio of red to green counters in a bag is $3 : 8$. If $15$ more red counters are added, the ratio becomes $3 : 4$. How many green counters are there?
▶ Show solution
Let one part be $x$: red $= 3x$, green $= 8x$.
$$\frac{3x + 15}{8x} = \frac{3}{4}$$
Cross-multiply: $4(3x + 15) = 3 \times 8x$
$12x + 60 = 24x \Rightarrow 60 = 12x \Rightarrow x = 5$
Green $= 8 \times 5 = \mathbf{40}$ counters.
Check: red was $15$, now $30$; $30 : 40 = 3 : 4$ ✓
Two brothers share sweets in the ratio $7 : 4$. The older brother gives $9$ sweets to the younger, and now they have equal amounts. How many sweets were there altogether?
▶ Show solution
Let one part be $x$: older $= 7x$, younger $= 4x$.
After the gift: $7x - 9 = 4x + 9$
$3x = 18 \Rightarrow x = 6$
Total $= 7x + 4x = 11x = 11 \times 6 = \mathbf{66}$ sweets.
Check: $42$ and $24$; after the gift, $33$ each ✓
$A$ is $(1,\ 2)$ and $B$ is $(13,\ 26)$. Find the coordinates of the point $P$ on $AB$ with $AP : PB = 2 : 1$.
▶ Show solution
Total parts $= 3$, so $P$ is $\dfrac{2}{3}$ of the way from $A$ to $B$.
$x$: change $= 13 - 1 = 12$; $\dfrac{2}{3} \times 12 = 8$; so $x_P = 1 + 8 = 9$.
$y$: change $= 26 - 2 = 24$; $\dfrac{2}{3} \times 24 = 16$; so $y_P = 2 + 16 = 18$.
$P = (9,\ 18)$
A currency graph is a straight line through the origin. The point $(200,\ 232)$ lies on it, where $x$ is pounds and $y$ is euros.
(a) Write the equation of the line. (b) Convert £$450$ into euros. (c) Convert €$696$ into pounds. (d) Write the ratio pounds : euros in the form $1 : n$.
▶ Show solution
(a) Gradient $= \dfrac{232}{200} = 1.16$, so $y = 1.16x$.
(b) $450 \times 1.16 = €522$
(c) $696 \div 1.16 = £600$
(d) $200 : 232 = 1 : 1.16$