Use a constant rate
Here A is an amount, q is its quantity and r is the amount per unit of quantity. The subscript “new” identifies the requested pair. Keep the rate’s units visible
- Unit rate Divide the known amount by the known quantity \(\displaystyle r=\frac{A}{q}\)
- New total Multiply the rate by the new quantity \(\displaystyle A_{\text{new}}=rq_{\text{new}}\)
- New quantity Divide a given total by the rate \(\displaystyle q_{\text{new}}=\frac{A_{\text{new}}}{r}\)
Worked example
A printer maintains a constant speed. Find the time for the new job
\[\begin{gathered}14\text{ posters take }9\text{ min}\\\text{New job: }42\text{ posters}\end{gathered}\]
- \(\displaystyle r=\frac{9}{14}\text{ min per poster}\)Use time per poster because the question asks for time
- \(\displaystyle t=\frac{9}{14}\times42\text{ min}\)Keep the exact rate while scaling
- \(\displaystyle t=27\text{ min}\)The job is three times as large, so the predicted time is three times as long. This assumes no change in printing speed
Common mistakes
Using the rate backwards
✕\(\displaystyle \text{Time}=\frac{14}{9}\times42\)✓\(\displaystyle \text{Time}=\frac{9}{14}\times42\)
Posters per minute and minutes per poster are reciprocal rates. Choose the direction needed by the calculationRounding too early
✕\(\displaystyle \text{Time}=0.64\times42=26.88\)✓\(\displaystyle \text{Time}=\frac{9}{14}\times42=27\)
A rounded intermediate rate changes the result. Keep the exact fraction until the endDifficulty levels
introScale a quantity forward at a simple constant rate
standardUse decimal rates or find a quantity from a total
challengeKeep an exact fractional rate, convert time units or combine two rates
Questions about this topic
How do I choose multiplication or division?+
Multiply a rate by a quantity to find a total. Divide the total by the rate to recover a quantity
Does a fixed fee allow direct proportion?+
A non-zero fixed fee means the total price is not directly proportional to the quantity
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, General Mathematics, chapter 1: Percentages and ratios
In this chapter
Convert between fractions, decimals and percentagesFind a percentage of a quantityExpress one quantity as a percentage of another1A Percentages: Find the original amount from a percentageIncrease or decrease a quantity by a percentageFind percentage changeAdd GST to a priceFind the GST and the pre-GST price from an inclusive priceExpress a ratio in simplest formComplete an equivalent ratioDivide a quantity in a given ratio1E Dividing in a ratio: Use one known part of a ratio to find the others1F Unitary method: Find the original amount from a percentage1F Unitary method: Use one known part of a ratio to find the othersScale a proportional quantity with the unitary methodDecide which option is the better value