Modelling with ratios

Ratios are not just an abstract idea; they are the maths behind maps, models, recipes and building materials. This unit brings together the ratio reasoning you have built and applies it to real measurement problems, through mathematical modelling. Whenever a real quantity is scaled, shared or mixed, a ratio is doing the work, and learning to model these situations turns ratios into a genuinely practical tool.

This year you learn to use mathematical modelling to solve practical problems involving ratios, setting up the problem, solving it, and interpreting the answer in its real context. The two most common situations are scaling, as in maps and models, and dividing a quantity in a given ratio, as in sharing or mixing.

Scale as a ratio

A scale is simply a ratio between a drawing and the real object. A map drawn at a scale of 1 to 100 means that every 1 centimetre on the map represents 100 centimetres in real life. So a line measuring 3 centimetres on the plan stands for 300 centimetres, which is 3 metres. To go from the drawing to reality you multiply by the scale; to go the other way you divide.

Scale works in both directions, which is what makes it so useful. A model car built at a scale of 1 to 20 has every length reduced to a twentieth of the real car, so a 4 metre car becomes a 0.2 metre model. Architects, mapmakers and engineers all rely on this ratio reasoning to represent huge things on a page or to plan small things precisely. Reading the scale as a ratio is the key to every such problem.

Dividing a quantity in a ratio

The other common model is sharing a measured quantity in a given ratio. The reliable method is to find the value of one part first. To share 600 millilitres in the ratio 2 to 3, add the parts to get 5 in total, then divide the quantity by that number: 600 divided by 5 is 120 millilitres per part. The two shares are 2 parts and 3 parts, giving 240 and 360 millilitres.

This one-part method handles a huge range of practical problems. Mixing concrete with cement and sand in the ratio 1 to 4 for a 25 kilogram batch means 5 parts in total, so one part is 5 kilograms, giving 5 kilograms of cement and 20 of sand. The modelling steps are always the same: read the situation as a ratio, find one part, scale up to each share, and then state the answer in the real units the question asked about. Setting a problem up, solving it, and interpreting the result in context is the heart of modelling, and ratios are one of its most powerful and everyday applications.

Teaching tip: real maps and model kits make scale tangible. Find the scale printed on a road map or a model box and work out a real distance or length together. Connecting the printed ratio to an actual measurement the student can picture makes scaling feel useful rather than abstract.

For dividing in a ratio, the find-one-part method is worth making automatic. Stress adding the parts first to get the total number of shares, since jumping straight to the fractions is where mistakes creep in. Cooking or mixing drinks at home gives plenty of natural practice.

Scaling a recipe

A recipe is a ratio in disguise. To make more, you multiply every ingredient by the same factor, which keeps the ratio between them fixed. Doubling the batch doubles the flour and the sugar together, so the taste stays the same while the quantities grow.

Reading distances off a map

A map scale lets you turn a length measured on paper into a real distance. At a scale of one to twenty-five thousand, every centimetre on the map is that many centimetres on the ground. Multiply the measured distance by the scale, then convert to kilometres to report it.

Dividing into more than two parts

The find-one-part method is not limited to two shares. For a three-part ratio you still add all the parts, divide the quantity by that total to get one part, then multiply for each share. Setting up the ratio and interpreting the shares in context is the modelling at work.