These questions ask you to find a missing quantity or judge whether two quantities have been scaled at one constant rate. Your job is to identify the matching pair, keep the order of the quantities fixed, and use multiplication rather than an unrelated addition.
Choose a direct-proportion method
First name the input and the dependent amount: for example, number of items and total mass. The known input is the base for any scale factor. Check that the setting promises a fixed rate, such as identical items or a constant price per metre. Then use one of the following forms. Keep units beside every number so that the division has a clear meaning.
| Situation | Calculation | Tiny example |
|---|---|---|
| Find an amount for a new input | Divide the known amount by the known input to get the amount per one, then multiply by the target input. | Eight identical tags use 52 centimetres of tape, so one uses 6.5 centimetres and eleven use 71.5 centimetres. |
| Use a scale factor directly | Compute target input divided by known input. Multiply the known amount by that factor. | Fourteen identical crates weigh 35 kilograms. For twenty-two crates the factor is 22 divided by 14, so the mass is 55 kilograms. |
| Find the input from a target amount | Find the amount per one, then divide the target amount by that unit rate. | Six metres cost 18 dollars, so the price is 3 dollars per metre and 33 dollars buys 11 metres. |
A quick check catches many errors. If the input is multiplied by a factor greater than one, the dependent amount must be multiplied by that same factor. For a reduction, both must shrink by the same factor. You can also divide the final amount by its input and compare that unit rate with the original one.
Reject the tempting wrong routes
| Tempting route | How to reject it |
|---|---|
| Add the increase in the item count straight to the old amount. | The two numbers usually have different units. Find what each added item contributes, or use the common multiplier. |
| Choose any result that rises when the input rises. | The direction alone is not enough. Divide amount by input for both pairs; the quotients must match. |
| Treat a difference between money amounts as a length. | A money difference cannot become metres. Convert through price per metre, with the unit written after each step. |
| Read an item count from the digits of a decimal measurement. | A decimal height is still a measurement. Divide total height by height per item instead of treating its digits as a count. |
Try it
Nine display panels need 27 identical bolts. At the same number of bolts per panel, how many bolts do 14 panels need? Enter the number of bolts.
A tap with a constant flow fills 3.6 litres in 2.4 minutes. How many litres does it fill in 5.8 minutes? Enter the number of litres.
Twelve gift boxes require 46 metres of ribbon when each box is decorated identically. How many metres of ribbon do 18 boxes require? Enter the number of metres.