0580 · 1.11Ratio and proportion

Understanding and simplifying ratios

0:00Two drinks can contain different amounts of liquid but still taste equally strong. What matters is the balance between concentrate and water, rather than the total amount. Ratio describes that balance, and proportion helps us preserve it when quantities change. By the end, you will use these ideas to share quantities, adapt recipes, interpret map scales and compare value. You need multiplication, division, fractions and simple unit conversions for this topic.

0:25A factor divides a number exactly. A common factor is shared by two or more numbers. Equivalent fractions express the same value using different numbers. Equivalent ratios work similarly, because multiplying or dividing every part by the same amount preserves the comparison. A ratio compares quantities in a stated order. Concentrate to water in the ratio two to three means two equal parts of concentrate for every three equal parts of water.

0:49Each part represents the same amount, such as one cup. Reversing the order gives water to concentrate as three to two, so labels matter. The whole drink contains five parts because 2 + 3 = 5. Concentrate therefore makes up two fifths of the mixture, not two thirds. The ratio two to three compares concentrate with water. The fraction two fifths compares concentrate with the whole drink, so these expressions answer different questions.

1:13Doubling both ingredients gives four parts of concentrate and six parts of water, without changing the balance. The multiplier connecting the original quantities to the new ones is called a scale factor. Multiplying just one ingredient changes the mixture instead. Equal ratios describe the same relative amounts, even when the actual quantities differ. An exam question might ask you to give twenty to thirty to forty in its simplest form.

1:37The highest common factor of all three numbers is ten. Dividing each part by ten gives two, three and four. So the simplest form is two to three to four. Simplest form normally uses whole numbers with no common factor greater than one across all the parts. You could divide by smaller common factors in several stages instead. Dividing each part by ten preserves every comparison because all quantities shrink together.

2:01Subtracting ten from every part would change those comparisons, so it does not produce an equivalent ratio. Consider the ratio zero point five to zero point seven five. Multiplying both parts by one hundred gives fifty to seventy-five. Their highest common factor is twenty-five. Dividing both numbers by twenty-five gives the simplest ratio, two to three. When comparing quantities of the same kind, convert them into the same units first.

2:25Take the ratio fifty centimetres to two metres. Since one metre is one hundred centimetres, the ratio becomes fifty to two hundred. Dividing both numbers by fifty gives one to four. The ratio is therefore one to four, not twenty-five to one.

Sharing amounts and scaling recipes

2:40Suppose Amira, Ben and Cara share one hundred and eighty pounds in the ratio two to three to four. Imagine the money divided into equal boxes, with each person receiving their stated number of boxes. The ratio tells us how many parts they receive, not their actual amounts. Finding the value of one part connects the ratio to the money. 2 + 3 + 4 = 9 parts altogether. 180 ÷ 9 gives twenty pounds per part.

3:05Amira receives two parts, which is forty pounds. Ben gets three parts, making sixty pounds. Cara has four parts, so her share is eighty pounds. 40 + 60 + 80 = 180 pounds, so the shares account for the whole amount. They also simplify back to two to three to four. Dividing by the number of people would give equal shares, which is not the requested ratio. We divide by the total number of parts because those parts have equal value.

3:29A recipe for four people uses three hundred grams of flour and two hundred millilitres of milk. To serve six people, the scale factor is 6 ÷ 4, which equals one point five. 300 × 1.5 gives four hundred and fifty grams of flour. For the milk, 200 × 1.5 is three hundred millilitres. This is proportional reasoning because every ingredient changes by the same factor as the number of servings.

3:54We assume each person receives the same serving size. Adding the same amount to every ingredient would not preserve the recipe. When scaling down, the same idea works with a multiplier smaller than one.

Map scales and value comparisons

4:05A map scale of one to fifty thousand compares a map length with its corresponding actual length. One centimetre on the map represents fifty thousand centimetres in reality. Both sides of the ratio use the same unit. The map is a smaller version of the real distances, so we multiply map lengths by fifty thousand. The distance between the school and library measures three point six centimetres on the map.

4:263.6 × 50000 gives one hundred and eighty thousand centimetres. Dividing by one hundred converts this to one thousand eight hundred metres. Dividing by one thousand gives one point eight kilometres. Keep the units attached to your working so the scale factor is interpreted correctly. Fifty thousand centimetres is not fifty thousand metres. If a question gives an actual distance and asks for its map length, divide by the scale factor instead.

4:50First express that actual distance in the unit required for the map measurement. Pack A contains seven hundred and fifty grams of cereal for three pounds. Pack B contains one point two kilograms for four pounds fifty. One point two kilograms equals one thousand two hundred grams. 1200 ÷ 750 gives a scale factor of one point six. Three pounds multiplied by one point six gives four pounds eighty. Pack B is better value because the same mass costs four pounds fifty instead.

5:18The smaller price alone does not identify better value because the packs contain different amounts. We compared what equal masses would cost, using proportional scaling. This comparison assumes the products have comparable quality. You could instead compare equal amounts of money and identify which option provides more cereal.

Using ratio differences and recap

5:36An exam question gives red to blue counters as three to five, with fourteen more blue counters than red counters. 5 − 3 = 2 parts, representing the difference. 14 ÷ 2 gives seven counters per part. Three parts of seven give twenty-one red counters, and five parts give thirty-five blue counters. 35 − 21 = 14, confirming the stated difference. Dividing fourteen by eight would be wrong because fourteen is not the total number of counters.

6:03When a total is given, add the ratio parts. When a difference is given, subtract the relevant parts to identify what that difference represents. These methods must work without a calculator on Paper two, although Paper four allows a calculator. Show the common factor, total parts or scale factor before writing your answer. Keep exact values during calculations and round only the final result when needed. Unless instructed otherwise, give answers that are not exact to three significant figures, and money to two decimal places.

6:33The main mistakes are changing only one ratio part, reversing the order and mixing units. Avoid them by labelling quantities and applying the same multiplier throughout. When sharing, identify whether the given amount represents the whole, one share or a difference. Your working should make that connection clear rather than treating every given amount as a total. Ratio describes relative amounts, while equivalent ratios preserve their balance through a common multiplier.

6:57Simplifying changes the numbers without changing that comparison. Sharing uses equal parts to connect a ratio with actual quantities. Recipes, map scales and value comparisons all use the same proportional reasoning, so understanding the relationship is more powerful than memorising separate procedures.