Express and simplify ratios, divide a quantity in a given ratio, work with rates, and solve direct proportion problems

What this dot point is asking

SEAB wants you to write and simplify ratios, divide a quantity in a given ratio, work with rates such as speed or price per item, and solve direct proportion problems where two quantities increase together. These ideas appear in money, measurement and travel-graph questions, so they are worth mastering early.

The answer

What a ratio is

A ratio compares two or more quantities of the same kind. The ratio $3 : 2$ means that for every $3$ of the first quantity there are $2$ of the second. Ratios have no units because the units cancel.

Simplifying a ratio

Divide every part by their common factor, just like simplifying a fraction. For example, $12 : 8$ divides by $4$ to give $3 : 2$. To compare unlike units, first convert to the same unit: $1\ \text{m} : 50\ \text{cm}$ becomes $100\ \text{cm} : 50\ \text{cm} = 2 : 1$.

Sharing in a ratio

To divide a quantity in a ratio:

1. Add the parts to find the total number of parts.
2. Divide the quantity by the total to find the value of one part.
3. Multiply by each share.

For example, to share \300$in the ratio$2 : 3 : 1$: there are$6$parts, one part is$\$50$, and the shares are \100$,$\$150$ and \50$.

Rates

A rate compares two quantities of different kinds, such as distance and time. Speed is the most common rate:

Other rates include price per kilogram and words typed per minute. A rate always carries units, for example km/h or \$/kg.

Direct proportion and the unitary method

Two quantities are in direct proportion when one is a constant multiple of the other: double one and the other doubles. The fastest way to solve such problems is the unitary method - find the value of one unit first, then scale up.

For example, if $4$ pens cost \6$, then one pen costs$\6 \div 4 = \1.50$, so$10$pens cost$10 \times \1.50 = \15$.

Examples in context

Example 1. A recipe scaled up. A recipe for $4$ people needs $200\ \text{g}$ of flour. For $6$ people, find one person's share first: $200 \div 4 = 50\ \text{g}$ each, so $6 \times 50 = 300\ \text{g}$. This unitary method works for any number of people and is the same idea as direct proportion.

Example 2. Comparing value. A $500\ \text{g}$ pack of rice costs \2.00$and a$2\ \text{kg}$pack costs$\$7.20$. Compare the rate per kilogram: the small pack is \4.00$/kg and the large pack is$\$3.60$/kg, so the large pack is better value. Turning each price into a rate makes the comparison fair.

Try this

• Cue. Simplify the ratio $18 : 24$. Divide both by $6$ to get $3 : 4$.
• Cue. Share \450$in the ratio$4 : 5$. There are$9$parts, one part is$\$50$, so the shares are \200$and$\$250$.
• Cue. If $5$ identical books cost \40$, find the cost of$8$books. One book is$\$8$, so $8$ books cost \64$.