Ratio and scale drawings

An Airfix Spitfire kit fits in your hand. The real aircraft's wings stretch more than eleven metres, about the length of a double-decker bus. Yet every panel line on the little model sits in the right place, and one small number printed on the box makes that possible.

That number is the scale, one to seventy-two.

Read that number the wrong way round, and your real Spitfire would have wings about two millimetres across.

Each video in this group of maths skills works without the others, and this one needs nothing before it.

A ratio compares two quantities in the same unit, written with a colon between them, and the order matters. One to five is not the same as five to one. A drawing scale is a ratio, and it is always drawing to real. At a scale of one to five, every millimetre on the drawing stands for five millimetres on the real object. That is a reduction, because the drawing is smaller. Two to one goes the other way: two millimetres of drawing for every one millimetre of the real part. That is an enlargement, used for small details. One to one is full size. Before any scale sum, decide whether the answer should be bigger or smaller than the number you start with. Then you know whether to multiply or divide. A working drawing of a bedside table is drawn at one to five, and its top measures eighty-four millimetres on the drawing. At one to five, how many millimetres long is the real table top? Four hundred and twenty. The real table is bigger, so you multiply by five. Now reverse it: five hundred and fifty millimetres tall. How tall on the drawing? A hundred and ten millimetres. The drawing is smaller, so you divide by five. Next, a twelve millimetre hinge in a detail view at two to one. How long on that view? Twenty-four millimetres. Two to one is an enlargement, so you multiply by two. A scale is chosen so the product fits the paper. A wardrobe two metres tall, drawn at one to ten, is two hundred millimetres on the drawing, which fits on an A4 sheet. Now the Spitfire. The small number on the box is its scale, one to seventy-two, and the model's wings span a hundred and fifty-six millimetres. To find the real wingspan, do you multiply or divide by seventy-two? Multiply. The real aircraft is bigger than the model, so the number must grow. A hundred and fifty-six times seventy-two is eleven thousand two hundred and thirty-two millimetres, about eleven point two metres. Divide instead, and you get about two millimetres, the toy-sized mistake.

Next, ratios between the parts of one thing. A square coaster has a top surface of eight thousand one hundred square millimetres. A walnut inlay covers two thousand two hundred and fifty of them, and the rest is birch. The question wants the ratio of walnut to birch, in its simplest form. One student writes eight thousand one hundred to two thousand two hundred and fifty, and simplifies it to eighteen to five. Eighteen to five, for walnut to birch: what has that ratio compared? The whole coaster with the walnut. Birch never appears in it, so it cannot be walnut to birch. So name the two parts in words first, walnut to birch, and find any part you are not given. Birch is the whole take away the walnut. Eight thousand one hundred minus two thousand two hundred and fifty is five thousand eight hundred and fifty square millimetres. Walnut to birch is two thousand two hundred and fifty to five thousand eight hundred and fifty. To simplify, divide both sides by the same number. Divide both by ten: two hundred and twenty-five to five hundred and eighty-five. Divide both by five: forty-five to a hundred and seventeen. Is forty-five to a hundred and seventeen in its simplest form? Not yet. Nine divides both, which gives five to thirteen. No whole number except one divides both five and thirteen, so five to thirteen is the simplest form. Check it by adding the parts: five plus thirteen is eighteen parts. Eight thousand one hundred divided by eighteen is four hundred and fifty a part, and five parts make two thousand two hundred and fifty, the walnut. One examiner's report, on a ratio question about coloured panels, says: A common misconception was to divide the total panel size, seven thousand two hundred square millimetres, by the area of six panels, two thousand four hundred square millimetres, giving an incorrect ratio of three to one. That is the whole-to-part slip from the coaster. The fix: name the two parts in words, subtract to find the missing one, and only then divide. Set out that way it tends to go well. A report on a different ratio question records: This question was well answered by most candidates where they gave a clearly laid out response to the ratio calculation.

Now sharing in a ratio. A jewellery maker, Priya, mixes clear casting resin with hardener in the ratio two to one by volume, and she needs a hundred and thirty-five millilitres of mixed resin. Picture the ratio as blocks: two blocks of resin and one block of hardener, all the same size. Is one block a hundred and thirty-five divided by two, or divided by three? By three. Count every block in the mix, and there are three parts. One part is a hundred and thirty-five divided by three, which is forty-five millilitres. Resin is two parts, ninety millilitres. Hardener is one part, forty-five millilitres. Check: ninety plus forty-five is a hundred and thirty-five, and ninety to forty-five simplifies to two to one. Now only seventy millilitres of hardener is left. What is the most mix Priya can make? Two hundred and ten millilitres. Hardener is one part, so one part is seventy, and the mix is three parts, three times seventy. That is a hundred and forty millilitres of resin with the seventy of hardener. Gear and pulley ratios use the same maths, and they have their own videos in the mechanisms chapter.

Four questions to close on, set on things this video has not used, with ratio and scale doing the work. What is twelve to eighteen in its simplest form? Two to three is the simplest form. Six divides both, and nothing but one divides two and three. Next, a shelf drawn at one to ten measures sixty millimetres. How long is the real shelf? Six hundred millimetres. The real shelf is bigger, so multiply by ten. Another: six hundred square millimetres, a hundred and fifty of it wasted. Used to waste? Three to one. Used is four hundred and fifty, so the ratio is four hundred and fifty to a hundred and fifty, which simplifies to three to one. Try this to finish: share sixty grams of pigment in the ratio three to two. Thirty-six grams and twenty-four grams. Five parts in all, twelve grams a part. And the Spitfire box: one to seventy-two, drawing to real. Ask bigger or smaller first, and the wings come out at eleven metres, not two millimetres.

Because these maths videos can be watched in any order, the thumbs up is how you keep score of the ones you can already do. When ratio and scale feel like you own this one, hit the thumb and leave it behind. If not, keep it among your saved videos and say the scale rule out loud once, and it sticks.

The next video in the chapter is Fractions and percentages: profit, waste and survey results.

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