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Know How to Work with Ratios on Numeracy Tests

A ratio is usually a pair of numbers with a colon between them, such as 2:1. It describes the relative size of two things – if you imagine two people splitting up a pile of sweets (or equal-valued banknotes, or anything else) and saying ‘two for me, one for you’, you’re imagining them dividing the loot in a 2:1 ratio – the first person gets two sweets for every one the other person gets.

Ratios are very similar to percentages, and you can solve them in exactly the same way.

If the question uses words such as ‘all together’ or ‘in total’, it’s almost certainly a total question – if not, then it’s most likely a normal ratio.

The Table of Joy is a technique for figuring out what sum you need to do when you have two amounts you know to be proportional – that is, if you double the size of one, you double the size of the other.

Normal ratio sums

A typical normal ratio question would ask something like ‘In a drink, juice and water are mixed in the ratio of 1:7. If you use 20 millilitres of juice, how much water do you need?’ Notice that there’s no mention of ‘all together’ or ‘total’.

To solve a regular ratio problem with the Table of Joy, here’s what you do:

  1. Draw out your Table of Joy grid.

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  2. Label the top with what you’re measuring – in this example, juice and water.

  3. Label the side with ‘ratio’ and ‘answer’.

  4. Fill in the ratio row. Here, you’d put 1 underneath ‘juice’ and 7 underneath ‘water’.

  5. Fill in the last row. In this case, 20 goes in the ‘juice’ column.

  6. Shade the grid like a chessboard, and write down the Table of Joy sum: multiply the two numbers on the same colour squares and divide by the other number. For this example, the sum is 7 x 20 ÷ 1.

  7. Do the sum. The answer here is 140 millilitres.

Total ratio sums

When your sum involves a total, there’s bad news and good news. The bad news is you need to think a little bit harder about setting up the sum; the good news is that the Table of Joy works just as well.

A question of this form might say, ‘Chris and Steve agree to split their lottery winnings in the ratio of 2:3. If they won £1,500 all together, how much would Chris get?’

Notice the key phrase ‘all together’ – that means you have to think about the total rather than just the numbers you’re given.

In this situation, you need to know the total of the ratio: 2 + 3 = 5. When it comes to writing your Table of Joy, remember that you need to include the total in the table, as well as the relevant part of the ratio – here, that’s Chris’s share, which is 2. You don’t care about Steve’s share at all, so you can safely ignore him.

To solve a total ratio problem with the Table of Joy, here’s what you do:

  1. Draw out your Table of Joy grid.

  2. Label the top with what you’re interested in – in this case, Chris’s share and the total.

  3. Label the side with ‘ratio’ and ‘answer’.

  4. Fill in the ratio row. Because Chris gets two shares out of a total of five, you put 2 under Chris and 5 under total.

  5. Fill in the last row. In this case, 1,500 goes in the ‘total’ column.

  6. Shade the grid like a chessboard, and write down the Table of Joy sum: multiply the two numbers on the same colour squares and divide by the other number. For this example, the sum is 1500 x 2 ÷ 5.

  7. Do the sum. The answer here is £600.

Three-part ratios

In some tests, you may see a ratio that looks like 2:3:4. This looks harder than a two-number ratio, but it really isn’t: you can do the sums in exactly the same way, by picking out the two interesting numbers in the ratio (or one number and the total, depending on the question) and putting them in the Table of Joy.

Simplifying ratios

Simplifying ratios is very similar to simplifying fractions. Just as you can write fractions in many different ways (you can write a half as 1/2 or 2/4 or 50/100, among others), you can write ratios in many different ways. The ratio 3:2 is the same as 6:4, or 15:10, or 60:40.

Just like with fractions, ratios have a simplest form, which just means a form where the two numbers have no factors (apart from 1) in common – there’s no number you can evenly divide both parts of the ratio by.

For instance, 3:2 is in its simplest form, because 3 and 2 have no factors other than 1 in common. 60:40 isn’t in its simplest form because you can divide both of the numbers by 20.

To simplify a ratio, here are the steps you take:

  1. Look for a number that divides evenly into both (or all) of the numbers in the ratio.

  2. If you can’t find one, you’re finished! The ratio is in its simplest form.

  3. If you found a suitable number in step 1, divide all of the numbers in the ratio by it.

  4. Go back to step 1 until you’re finished.

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