Ratio Problems

A ratio problem is a longer ratio question with more than one step. You still find the value of one part, but first you may need to combine two ratios, take a fraction off the total, check the ingredients of a mixture, or follow a change.

In the exam these questions come in the second half of the paper, for 3 to 5 marks. They may ask for a number of things, a fraction, a percentage, or whether someone's claim is correct.

Write down what each number stands for as you go, such as "red: 24" or "one part: 6". Many marks are lost by using the right method on the wrong amount.

Every ratio problem turns on the value of one part. Before you divide, decide what the amount you know stands for.

• A total is all the parts: divide by the sum of the ratio numbers.
• One share is only that person's parts: divide by that person's number.
• A difference between two shares: divide by the gap between their numbers.

With a three-part ratio, the total is all three numbers added. In 2:5:4 there are 2+5+4=11 parts, even if the question only talks about two of the amounts.

Some questions give two ratios that share an amount, such as tea : coffee =3:4 and coffee : juice =2:5. Coffee is in both, but as 4 in one and 2 in the other, so you can't just write the numbers side by side.

Make the shared amount the same number in both ratios. Use a common multiple, here 4. The first ratio stays 3:4. Multiply the second by 2: coffee : juice =4:10. So tea : coffee : juice =3:4:10

Now use the three-part ratio as usual. There are 3+4+10=17 parts, so coffee is \frac{4}{17} of the drinks.

Ratios are sometimes hidden in words. "For every 5 boys there are 6 girls" means boys : girls =5:6. "There are twice as many apples as pears" means apples : pears =2:1. "There are three quarters as many apples as pears" means apples : pears =3:4

Some questions take a fraction or a percentage of the total first. Only the rest is shared in the ratio.

For example, there are 80 counters. \frac{1}{4} of them are red and the rest are blue and green in the ratio 2:3. Red is 80\div4=20, so 80-20=60 counters are shared. One part is 60\div5=12, so there are 24 blue and 36 green.

To give an answer as a fraction or percentage of the whole, divide by the original total, not by the amount that was shared. In the example, green is \frac{36}{80} of all the counters, which is 45\%

A mixture in a ratio tells you how much of each thing goes in. For squash : water =1:4, a 750 ml drink has 5 parts of 150 ml: 150 ml of squash and 600 ml of water.

To decide whether there is enough of everything, work out how much of each you need. Then compare each amount with what you have.

To find the greatest amount you can make, divide what you have of each ingredient by its number of parts. The smallest answer is the most that one part can be, because that ingredient runs out first. Multiply it by the total number of parts.

If you know the actual numbers, change them first and then write the new ratio. With 20 red and 30 blue counters, take away 5 red: red : blue =15:30=1:2

If you are given a ratio and a total, work out the actual amounts first. Then make the change to the amounts, never to the numbers in the ratio. Anything that is not added or taken away stays the same.

When things only move from one person to another, the total does not change. Use the total to find the value of one part in the new ratio.

• Writing two ratios side by side without matching them. For red : blue =2:3 and blue : green =1:4, the answer is not 2:3:4 or 2:4:4. Make blue the same number in both first: 2:3:12
• Adding all four numbers from two ratios. Combine the ratios first, then add the three numbers in the combined ratio.
• Sharing one person's share as if it were the total. If you are told one share, divide it by that person's number of parts.
• Sharing the whole amount when a fraction has been taken off. Take the fraction or percentage off first, then share only the rest.
• Checking only the total amount of a mixture. Work out how much of each ingredient the ratio needs, and check each one.
• Finding a percentage of the wrong amount. Divide by the total the question asks about, usually the original total.