FR1, FR12, HR1, HR12 - Unit rates and compound measures

Rate and unit rate

A rate compares quantities with different units. A unit rate gives the amount for one unit of the other quantity. “£$2.40$ per kilogram” means each kilogram costs £$2.40$; “£$12$ per hour” means each paid hour earns £$12$.

Worked example 1: comparing different masses

Question. The same rice is sold as $750\,\mathrm g$ for £$1.80$ or $1.2\,\mathrm{kg}$ for £$3.12$. Which has the lower unit price?

Plan. Convert both masses to kilograms, then compare pounds per kilogram.

Answer. The $750\,\mathrm g$ pack is better value: £$2.40/\mathrm{kg}$ compared with £$2.60/\mathrm{kg}$.

Check. $6\,\mathrm{kg}$ would be eight small packs costing £$14.40$, or five large packs costing £$15.60$, confirming the comparison.

Worked example 2: comparing capacities

Question. The same detergent costs £$2.70$ for $750\,\mathrm{ml}$, or £$6.80$ for $2$ litres. Compare the unit prices.

Plan. Express both capacities in litres and divide cost by litres.

Answer. The $2$-litre pack is better value: £$3.40$ per litre rather than £$3.60$ per litre.

Check. At the smaller pack's rate, $2$ litres would cost $2\times3.60=$ £$7.20$, more than £$6.80$.

Worked example 3: pay for part of an hour

Question. A worker earns £$12.40$ per hour and works $3$ hours $45$ minutes, all paid at that rate. Find the pay.

Plan. Convert $45$ minutes into a fraction of an hour.

Answer. £$46.50$.

Check. Three hours earns £$37.20$; three quarters of an hour earns £$9.30$. Their sum is £$46.50$.

Worked example 4: overtime and an unpaid break

Question. A shift runs from $08{:}00$ to $18{:}00$, including a $30$-minute unpaid break. The first $8$ paid hours earn £$12$ per hour; later time earns one and a half times that rate. Find total pay.

Plan. Remove unpaid time, then separate normal and overtime hours.

The shift lasts $10$ hours, so paid time is $9.5$ hours. Overtime is $9.5-8=1.5$ hours at $1.5\times12=$ £$18$ per hour.

Answer. £$123$.

Check. Normal-rate pay for all $9.5$ hours would be £$114$. The overtime premium adds $1.5\times6=$ £$9$, giving £$123$.

Watch out

$3$ hours $45$ minutes is $3.75$ hours, not $3.45$. “Per hour” tells you the unit needed before multiplying.

Distance, speed and time

Use consistent units: kilometres with hours gives kilometres per hour. These relationships are methods to know, not supplied formulas on the formula sheet.