To divide a quantity in a given ratio, add the parts to find the total number of shares, divide the quantity by that total to find one share, then multiply each part by one share. For a map scale written as 1 : n, multiply any map length by n to get the real length in the same unit, then convert units as needed. Areas do not scale the same way as lengths: because area is length times length, the area scale factor is the length scale factor squared, so a map scale of 1 : n gives an area scale of 1 : n².
Ratio division itself is usually fine. Students lose marks on two different mistakes that sit right next to each other in the syllabus: mixing up which part of the ratio goes with which quantity, and forgetting that map scales behave differently for length, area, and volume.
The length scale factor tells you how map distances relate to real distances. But when the question asks about a map area, that same factor cannot be used directly: it has to be squared first. A student will often work out the length scale factor correctly, then multiply the map area by it instead of by its square, and lose marks on an otherwise complete answer.
Ratio division
Total shares = sum of ratio parts. One share = quantity ÷ total shares.
Map length
Real length = map length × n, where scale is 1 : n
Map area
Real area = map area × n²
Reverse (real to map)
Map length = real length ÷ n. Map area = real area ÷ n²
A sum of $360 is divided between Amira and Bala in the ratio 4 : 5. Find each person's share.
Solution
Quick check: the two shares should always add back to the original quantity. If they don't, you likely divided by the wrong total or swapped a ratio part.
A map has a scale of 1 : 25 000. The distance between two towns on the map is 8 cm. Find the actual distance in kilometres.
Solution
Using the same map, scale 1 : 25 000, a park has an area of 3 cm² on the map. Find its actual area in km².
Solution
The step students get wrong
Multiplying the map area directly by the length scale factor n, instead of by n². This gives an answer that is far too small. Whenever the quantity is an area, square the scale factor first, before multiplying. The same rule extends to volume in solid figures: the volume scale factor is n³, the cube of the length scale factor.
A map has a scale of 1 : 10 000. A field has an actual area of 4 km². Find its area on the map in cm².
Solution
Exam tip: convert units before applying the scale factor, or straight after, whichever keeps the numbers smaller. Either order works as long as you are consistent. Mixing units partway through a calculation is the most common source of a wrong final answer.
Why is the area scale factor squared and not the same as the length scale factor?
Area is measured in square units, made from two lengths multiplied together, such as length times width. If every length on a shape is scaled by n, then an area made of two such lengths is scaled by n × n = n². This is the same reason a similar figure's area scale factor equals the square of its length scale factor, which you can see worked through in the similarity tests guide.
Does the same squaring rule apply to volume?
Volume is made of three lengths multiplied together, so the volume scale factor is n³, the cube of the length scale factor. This matters for 3D map or model questions, such as converting a scale model's volume to the actual volume of a building. See the fuller treatment in the area and volume scale factor guide.
How do I know whether a question wants a ratio, a rate, or a scale?
A ratio compares two quantities of the same kind, such as money to money. A map scale is a special ratio comparing map length to real length, always written as 1 : n. If the question gives you two quantities with different units, such as distance and time, that is a rate question instead, not a ratio question. A map scale is also an example of direct proportion: real length rises in step with map length, always in the same fixed ratio.
What if the ratio has three parts instead of two?
The method is identical, add all three parts to find the total shares, divide the quantity by the total to find one share, then multiply each part by one share. For example, dividing $600 in the ratio 2 : 3 : 5 gives total shares of 10, one share of $60, and portions of $120, $180, and $300.
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