Ratios and Rates
A ratio compares quantities of the same kind, while a rate compares different kinds, such as kilometres per hour. This NSW Year 9 Advanced maths subtopic revises Stage 4 ratios and rates (NESA outcome MA4‑RAT‑C‑01) and adds multi-step problems.
You will simplify ratios with mixed units, share amounts in a given ratio, and use rates to find costs, wages, distances and populations. You will combine two ratios into \(a : b : c\), compare best buys, use a map scale with a cycling speed, find an average speed over two legs, solve a marble problem where the ratio changes, and find how long two taps and a drain take to fill a tank. It is part of Year 9 Advanced Working with Numbers.
Theory
A ratio compares quantities in the same units; a rate compares quantities in different units, such as km/h or \(\$\)/kg. This Year 9 Advanced maths page revises the skills from earlier years in full, then goes further: sharing from a difference, joining two ratios, ratios that change, average speed and work rates.
A ratio such as \(3 : 5\) compares two or more quantities measured in the same units. Change both to the same unit first, then drop the units: \(1.5\text{ m} : 60\text{ cm} = 150 : 60 = 5 : 2\).
A ratio is in simplest form when its terms are whole numbers with no common factor. Divide every term by the highest common factor (HCF). Order matters: \(5 : 2\) is not the same as \(2 : 5\).
A ratio splits a quantity into equal parts. In \(3 : 5\) there are \(8\) parts altogether, the first share is \(\dfrac{3}{8}\) of the whole, and the shares differ by \(2\) parts. The unitary method finds the value of \(1\) part first.
A rate compares quantities in different units, such as \(72\) km/h or \(\$3.40\)/kg. A unit rate is the amount for one unit of the second quantity. A rate can also be given per \(100\) or per \(1000\), such as a birth rate of \(12\) per \(1000\) people.
Sharing a quantity in the ratio \(a : b\):
Speed, distance and time:
How to solve a ratio problem
- Same units: change every quantity to one unit, then simplify by the HCF. Clear decimals by multiplying every term by \(10\) or \(100\).
- Match the information to parts: a total matches all the parts, one share matches its own parts, a difference matches the difference in parts.
- Find 1 part, then multiply up to the share you want. Check that the shares add to the total.
- Join two ratios by making the shared term equal with the lowest common multiple.
- A changing ratio: call the shares \(ak\) and \(bk\), keep the unchanged quantity, write the new ratio as a fraction equation and solve for \(k\).
- Money in a ratio after a fixed amount: take any fixed payment out first, share the rest, then add the fixed amount back to that person.
How to solve a rate problem
- Find the unit rate by dividing: \(\$21.60\) for \(8\) kg is \(\$2.70\)/kg. Multiply the unit rate to scale up; divide by it to find how many units.
- Best buy: compare the cost of the same amount, such as cents per \(100\) g. Lower cost per unit is the better buy.
- Per 100 or per 1000: write the rate as a fraction and set up an equation, such as \(\dfrac{\text{births}}{\text{population}} = \dfrac{12}{1000}\).
- Map scale then speed: scale up the map distance, convert it to km, then use \(T = \dfrac{D}{S}\).
- Times: \(0.4\) h is \(0.4 \times 60 = 24\) min, and \(0.25\) min is \(15\) s. Count clock times carefully: \(9\) am to \(2\) pm is \(5\) hours.
The \(\$150\) is a difference, so match it to the difference in parts, not to the total.
| \(7 - 2\) | \(=\) | \(5\text{ parts}\) |
| \(5\text{ parts}\) | \(=\) | \(\$150\) |
| \(1\text{ part}\) | \(=\) | \(\$150 \div 5 = \$30\) |
| \(\text{Total parts}\) | \(=\) | \(2 + 3 + 7 = 12\) |
| \(\text{Total}\) | \(=\) | \(12 \times \$30\) |
| \(\) | \(=\) | \(\mathbf{\$360}\) |
Check: Ana gets \(\$60\), Ben \(\$90\) and Cara \(\$210\). These add to \(\$360\), and \(210 - 60 = 150\).
Both ratios contain \(y\). Make the \(y\) parts equal, using the lowest common multiple of \(4\) and \(6\), which is \(12\).
| \(x : y\) | \(=\) | \(3 : 4 = 9 : 12\quad(\times 3)\) |
| \(y : z\) | \(=\) | \(6 : 5 = 12 : 10\quad(\times 2)\) |
| \(x : y : z\) | \(=\) | \(\mathbf{9 : 12 : 10}\) |
\(9\), \(12\) and \(10\) have no common factor, so the ratio is already in simplest form.
The number of girls does not change. Call the starting numbers \(4k\) boys and \(7k\) girls, then write the new ratio as a fraction.
| \(\dfrac{4k + 6}{7k}\) | \(=\) | \(\dfrac{2}{3}\) |
| \(3(4k + 6)\) | \(=\) | \(2 \times 7k\) |
| \(12k + 18\) | \(=\) | \(14k\) |
| \(18\) | \(=\) | \(2k\) |
| \(k\) | \(=\) | \(9\) |
| \(\text{Girls} = 7k\) | \(=\) | \(\mathbf{63}\) |
Check: boys go from \(36\) to \(42\), and \(42 : 63 = 2 : 3\).
Work with the fraction of the order done in one hour. A does \(\dfrac{1}{6}\) of the order each hour and B does \(\dfrac{1}{4}\).
| \(\text{A in 1 h}\) | \(=\) | \(\dfrac{1}{6}\) |
| \(\text{Left}\) | \(=\) | \(1 - \dfrac{1}{6} = \dfrac{5}{6}\) |
| \(\text{Rate together}\) | \(=\) | \(\dfrac{1}{6} + \dfrac{1}{4} = \dfrac{2}{12} + \dfrac{3}{12}\) |
| \(\) | \(=\) | \(\dfrac{5}{12}\text{ per hour}\) |
| \(\text{Time together}\) | \(=\) | \(\dfrac{5}{6} \div \dfrac{5}{12}\) |
| \(\) | \(=\) | \(\dfrac{5}{6} \times \dfrac{12}{5} = 2\text{ h}\) |
| \(\text{Total time}\) | \(=\) | \(1 + 2 = \mathbf{3}\text{ hours}\) |
Time \(=\) work left \(\div\) rate. A tap that empties a tank works against the others, so its rate is subtracted.
Common pitfalls
Frequently asked questions
What is the difference between a ratio and a rate?
A ratio compares quantities measured in the same units, such as 3 cups of flour to 2 cups of water, so it has no units. A rate compares quantities in different units, such as 80 kilometres per hour or 4 dollars per kilogram.
How do you share an amount in a ratio?
Add the parts of the ratio, divide the amount by the total number of parts to find one part, then multiply by each share's parts. For \(\$120\) in the ratio \(1 : 3\) there are \(4\) parts, one part is \(\$30\), so the shares are \(\$30\) and \(\$90\).
How do you solve a ratio problem when you know the difference?
Subtract the parts of the ratio to get the difference in parts, and divide the given difference by it to find one part. Then multiply to get each share or the total.
How do you combine two ratios into one?
Find the term the two ratios share and make it equal in both, using the lowest common multiple. For \(a : b = 1 : 2\) and \(b : c = 3 : 4\), make \(b\) equal \(6\): \(a : b : c = 3 : 6 : 8\).
How do you find average speed for a whole trip?
Divide the total distance by the total time. Find the time for each part of the trip first with time equals distance divided by speed. Do not just average the two speeds.
How do you work out how long two taps take to fill a tank together?
Write each tap's rate as a fraction of the tank per hour, such as \(\dfrac{1}{3}\) for a tap that takes \(3\) hours. Add the rates, subtract any drain, then divide \(1\) tank by the combined rate.