Number and algebra · Lesson 6 of 7
Rates and best buys
A price tag tells you almost nothing until you know how much you get for it. Divide, make the amounts match, then compare.
Change the prices, sizes and units. The scales weigh each option by its unit price, so the dearer side sinks and the better buy rises.
Option A
Option B
Same amount on both sides
The default assumption is that "cheaper" means the lower price tag. Instead, a fair comparison needs the same amount of the thing on both sides. A 500 g bag for $4.20 and a 1.2 kg bag for $9.60 are different amounts, so their tags cannot be compared directly.
A rate is a comparison of two different kinds of quantity: dollars and grams, kilometres and hours, pages and minutes. The move is always the same: divide to find the amount for one unit. Dividing $4.20 by 500 shares the price equally across every gram, which gives the cost per gram, and scaling that up gives the cost per 100 g or per kilogram.
Once both options are priced per the same unit, the comparison is just two numbers side by side. The practical consequence is three formulas that are really one idea: unit price is price ÷ quantity, speed is distance ÷ time, and fuel consumption in Australia is litres ÷ kilometres × 100, read as litres per 100 km.
A rate can be read in either direction. 90 km in 1 hour is 90 km per hour, and it is also 1/90 of an hour per kilometre. Pick the direction the question needs: "how far" wants kilometres per hour, "how long" wants time per kilometre.
The seven rate methods
Every rate question on the test is one of these seven recipes. Each is a short, fixed sequence, and all of them start with the same step: divide to find the amount for one unit.
1. Unit price
- Divide the price by the quantity. That is the price for one unit.
- Supermarket labels use "per 100 g" or "per 100 mL", so scale to 100 units if it helps: divide the quantity by 100 first, then divide the price by that.
$4.20 for 500 g → 500 ÷ 100 = 5 lots of 100 g → $4.20 ÷ 5 = $0.84 per 100 g
2. Comparing when units differ
- Convert both quantities to the same unit (1 kg = 1000 g, 1 L = 1000 mL).
- Work out each unit price on the same basis, then compare. Lower is the better buy.
1.2 kg = 1200 g → $9.60 ÷ 12 = $0.80 per 100 g, which beats $0.84
3. Speed, distance, time
- speed = distance ÷ time; distance = speed × time; time = distance ÷ speed.
- Convert minutes to decimal hours before you divide: minutes ÷ 60. 45 min = 0.75 h and 20 min = 1/3 h. This is the biggest error source.
126 km in 1 h 45 min → 45 ÷ 60 = 0.75 → 126 ÷ 1.75 = 72 km/h
4. Fuel consumption and trip cost
- Litres per 100 km = litres used ÷ km driven × 100. Lower is better.
- Litres for a trip = L/100 km × km ÷ 100.
- Cost = litres × price per litre.
7.5 L/100 km, 240 km, $1.95/L → 7.5 × 240 ÷ 100 = 18 L → 18 × $1.95 = $35.10
5. Rates of pay and other "per" rates
- Earnings = rate per hour × hours (turn minutes into decimal hours first).
- Any "per" rate works the same way: pages per minute × minutes = pages; students ÷ teachers = students per teacher.
$62.50 per hour for 5 h 20 min → 20 ÷ 60 = 0.333… → 62.50 × 5.333… = $333.33
6. Converting a rate
- km/h to m/s: × 1000 (km to m) then ÷ 3600 (hours to seconds). Together that is ÷ 3.6.
- m/s to km/h: × 3.6.
- Per minute to per hour: × 60. Per hour to per minute: ÷ 60.
90 km/h → 90 × 1000 ÷ 3600 = 25 m/s; 40 pages per minute → 40 × 60 = 2400 per hour
7. Multipacks and "buy N get M free"
- Count what you pay and count what you receive.
- Effective price per item = total paid ÷ items received.
Buy 2 get 1 free at $3.00 each → pay $6.00, receive 3 → $2.00 per item
Rate solver that shows its working
Pick a problem type and fill in the numbers
Every step is shown with its reason, not just the answer. Change a number and the working updates.
Reading the per 100 g label
Australian supermarket shelf labels print a unit price in small type: per 100 g, per 100 mL, per litre or per item. The label does the division for you, and the test often expects you to do the same division yourself.
Bigger packs are usually cheaper per 100 g, but not always, and a question can be built around the exception. The only safe move is to compute each unit price and look at the numbers.
Conversions to know by heart
These save time in both sections of the test. The minutes table matters most: nearly every speed and pay question hides a minutes-to-hours step.
Minutes to decimal hours
| Minutes | Decimal hours | As a fraction of an hour |
|---|---|---|
| 5 min | 0.083 | 1/12 |
| 10 min | 0.167 | 1/6 |
| 15 min | 0.25 | 1/4 |
| 20 min | 0.333 | 1/3 |
| 30 min | 0.5 | 1/2 |
| 40 min | 0.667 | 2/3 |
| 45 min | 0.75 | 3/4 |
| 50 min | 0.833 | 5/6 |
Speed conversions
| km/h | m/s | Working |
|---|---|---|
| 36 km/h | 10 m/s | 36 000 m ÷ 3600 s |
| 60 km/h | 16.7 m/s | 60 000 ÷ 3600, rounded |
| 90 km/h | 25 m/s | 90 000 ÷ 3600 |
| 100 km/h | 27.8 m/s | 100 000 ÷ 3600, rounded |
Unit-price anchors
| If you know | Then | Why |
|---|---|---|
| Price per 100 g | Price per kg = × 10 | There are ten lots of 100 g in a kilogram |
| Price per 100 mL | Price per litre = × 10 | There are ten lots of 100 mL in a litre |
| Price per kg | Price per 500 g = ÷ 2; per 250 g = ÷ 4 | 500 g is half a kilogram, 250 g is a quarter |
Hide a column, say each answer aloud, then click a striped cell to check it.
Checkout challenge
Ten rounds. Most rounds show two pack options: click the better buy. Every third round is a speed or distance item instead, and you type the answer. Score and best streak are tracked.
Round 1 of 10
Common traps
- Comparing price tags, not unit prices. $9.60 is more than $4.20, but it buys more than twice as much. Divide first, then compare.
- Dividing the wrong way. Quantity ÷ price gives grams per dollar, which is a real rate but not the one the question asks for. Price ÷ quantity gives dollars per gram. Check that the units of the answer match the question.
- Mixing units. Comparing 400 g with 1.5 kg as if both were "grams" gives a nonsense answer, and so does writing 1 h 30 min as 1.3 h. Convert to one unit before any division: 1.5 kg = 1500 g, 1 h 30 min = 1.5 h.
- Trusting "per 100 g" as a serving. The shelf label compares price per 100 g of what is in the pack. It says nothing about how much you use or eat, so a concentrate can look dear per 100 mL and still be the better buy per wash.
- Assuming the bigger pack is always cheaper. Usually true, but a question can be built around the exception. Compute both unit prices every time.
- Averaging two speeds. Going out at 60 km/h and back at 40 km/h does not average 50 km/h, because the slower leg takes longer. Average speed is always total distance ÷ total time.
- Reading L/100 km upside-down. It is litres used, per 100 km, so a lower number means less fuel. A car at 6 L/100 km is more economical than one at 9 L/100 km.
ACER-style practice
Eight questions in the formats the real test uses: multiple choice, type your answer, and true-or-false tables. Questions marked "no calculator" belong to the test's final section, where you cannot use one. Check each answer for a full explanation.
What to remember
- A rate compares two different kinds of quantity. Divide to get the amount per one unit, then compare on the same unit.
- Unit price:
price ÷ quantity. Lower is the better buy. Convert kg to g or L to mL before dividing. - Speed:
distance ÷ time; distance:speed × time; time:distance ÷ speed. - Minutes to hours:
÷ 60. 45 min = 0.75 h, 20 min = 0.333 h, 1 h 30 min = 1.5 h. - Fuel:
L/100 km × km ÷ 100gives litres; multiply by the price per litre for the cost. Lower L/100 km is better. - km/h to m/s:
÷ 3.6. Per minute to per hour:× 60. - Deals: effective price per item =
total paid ÷ items received.
All questions on this page are original and are not reproduced from ACER material. LANTITE is administered by the Australian Council for Educational Research.
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