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Fractions, decimals and percentages

Fractions, Decimals and Percentages

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Pizzas are very useful, mathematically speaking. However much we hate fractions, we all know what half a pizza looks like, and that’s the point. Numbers don’t have any intrinsic meaning, and we can’t picture them unless they relate to something in the real world, so pizzas are just a useful way of illustrating fractions, decimals and percentages.

They all do the same job of showing what share of something you have, and a common question involves converting from one to another, so here are a few tips…

Fractions to Decimals

Calculator

  • Simply divide the numerator by the denominator, eg 3/4 = 3 ÷ 4 = 0.75.

Non-calculator

You can always use the standard ‘bus stop’ method to divide the numerator by the denominator on paper (or in your head), but the numbers may be easy enough for you to use a shortcut.

  • If the denominator is a power of 10 (eg 10 or 100), write the numerator down straight away as a decimal. You just have to make sure you end up with the digits in the right columns, eg a fraction involving hundredths needs to end in the second column after the decimal point, so 29/100 = 0.29.
  • If the denominator ends in zero, you may be able to simplify the fraction into tenths first and then convert it into a decimal, eg 16/20 = 8/10 = 0.8.
  • If you express the fraction in its lowest terms by simplifying it (ie dividing the numerator and denominator by the same numbers until you can’t go any further), you may  recognise a common fraction that you can easily convert, eg 36/45 = 4/5 = 0.8. Just make sure that you learn all these by heart, especially the eighths!

Quiz

  1. What is 5/10 as a decimal?
  2. What is 8/40 as a decimal?
  3. What is 36/60 as a decimal?
  4. What is 27/36 as a decimal?
  5. What is 77/88 as a decimal?

Fractions to Percentages

Calculator

  • Simply divide the numerator by the denominator, multiply by 100 and add the ‘%’ sign, eg 3/4 = 3 ÷ 4 x 100 = 0.75 x 100 = 75%.

Non-calculator

You can always convert the fraction into a decimal (see above) and then multiply by 100 and add the ‘%’ sign. Otherwise, try these short cuts in order.

  • If the denominator is a factor of 100 (eg 10, 20, 25 or 50), multiply the numerator by whatever number will turn the denominator into 100 and add the ‘%’ sign, eg 18/25 = 18 x 4 = 72%.
  • If the denominator is a multiple of 10 (eg 30, 40 or 70), divide the numerator by the first digit(s) of the denominator to turn the fraction into tenths, multiply the numerator by 10 and add the ‘%’ sign, eg 32/80 = 32 ÷ 8 x 10 = 4 x 10 = 40%.
  • If you express the fraction in its lowest terms by simplifying it (ie dividing the numerator and denominator by the same numbers until you can’t go any further), you may  recognise a common fraction that you can easily convert from memory, eg 8/64 = 1/8 = 12.5%.

Quiz

  1. What is 4/10 as a percentage?
  2. What is 6/20 as a percentage?
  3. What is 24/40 as a percentage?
  4. What is 14/70 as a percentage?
  5. What is 40/64 as a percentage?

Decimals to Fractions

Every decimal is really a fraction in disguise, so the method is the same whether you’re allowed a calculator or not.

Calculator/non-calculator

  • Check the final column of the decimal (eg tenths or hundredths) and place all the digits over the relevant power of 10 (eg 100 or 1000) before simplifying if necessary, eg 0.625 = 625/1000 = 5/8.

Quiz

  1. What is 0.4 as a fraction?
  2. What is 0.25 as a fraction?
  3. What is 0.24 as a fraction?
  4. What is 0.875 as a fraction?
  5. What is 0.375 as a fraction?

Decimals to Percentages

Again, this is an easy one, so the method is the same whether you’re allowed a calculator or not.

Calculator/non-calculator

  • Multiply by 100 and add the ‘%’ sign, eg 0.375 x 100 = 37.5%.

Quiz

  1. What is 0.27 as a percentage?
  2. What is 0.1 as a percentage?
  3. What is 0.55 as a percentage?
  4. What is 0.001 as a percentage?
  5. What is 1.5 as a percentage?

Percentages to Fractions

You can think of a percentage as simply a fraction over 100, so the method is easy enough whether you’re allowed a calculator or not.

Calculator/non-calculator

  • If the percentage is a whole number, remove the ‘%’ sign, place the percentage over 100 and simplify if necessary, eg 75% = 75/100 = 3/4.
  • If not, turn the fraction into a whole number as you go by multiplying the numerator and denominator by whatever number you need to (usually 2, 3 or 4), eg 37.5% = (37.5 x 2) / (100 x 2) = 75/200 = 3/8.

Quiz

  1. What is 22% as a fraction?
  2. What is 15% as a fraction?
  3. What is 37.5% as a fraction?
  4. What is 87.5% as a fraction?
  5. What is 6.25% as a fraction?

Percentages to Decimals

This is easy enough, so the method is the same whether you’re allowed a calculator or not.

Calculator/non-calculator

  • Remove the ‘%’ sign and divide by 100, eg 70% ÷ 100 = 0.7.

Quiz

  1. What is 40% as a decimal?
  2. What is 70% as a decimal?
  3. What is 35% as a decimal?
  4. What is 45.5% as a decimal?
  5. What is 62.1% as a decimal?

Ordering Fractions, Decimals and Percentages

A common question in the 11+ or 13+ involves putting a list of fractions, decimals and/or percentages in size order—either from largest to smallest or smallest to largest.

There are a number of ways of doing this, and it depends what kind of numbers are involved. However, a good first step is to start with the first two numbers and ask yourself if one is ‘obviously’ bigger than another. For instance, it might be quite difficult to compare 1/17 and 18/19 by converting them to fractions with the same denominator, but you don’t have to because 1/17 is clearly smaller!

After that, you can look at each number one by one and work out where it fits in your list. To keep track of everything, it’s a good idea to put numbers in pencil next to each value. Once you have the final order, you can write them all down on the answer line.

One simple question you can always ask yourself is whether the two fractions, decimals or percentages are smaller or larger than a half. If one is smaller but the other is larger, then the answer’s obvious.

If that doesn’t work, here are a few more ways to do it.

Ordering Fractions

If two fractions have the same denominator, the larger one will be the one with the larger numerator, eg 2/3 is bigger than 1/3.

If the fractions have different denominators, turn them into fractions with the same denominator and then compare the numerators, eg 5/6 and 7/8 are the same as 40/48 and 42/48, so 7/8 must be larger.

Quiz

  1. Put these numbers in order from largest to smallest: 1/2, 1/4, 2/5, 4/7, 5/8
  2. Put these numbers in order from largest to smallest: 3/4, 1/8, 5/6, 4/9, 3/8
  3. Put these numbers in order from largest to smallest: 4/5, 1/9, 3/4, 7/8, 1/4
  4. Put these numbers in order from largest to smallest: 1/3, 3/4, 2/3, 1/8, 5/6
  5. Put these numbers in order from largest to smallest: 2/5, 1/2, 2/3, 4/5, 3/4

Ordering Decimals

Decimals are easy to sort. It’s a bit like putting words in alphabetical order:

  • Start with the first digit after the decimal point, which is the number of tenths. The number with the bigger first digit is bigger overall, eg 0.2 is bigger than 0.1.
  • If the numbers have the same number of tenths, compare the hundredths, eg 0.12 is bigger than 0.11.
  • Repeat until you find the first digit that’s different. Just remember that if one number ends before you get a different number, it will always be smaller, eg 0.45 is smaller than 0.456.

Quiz

  1. Put these numbers in order from smallest to largest: 0.2, 0.3, 0.11, 0.2, 0.33
  2. Put these numbers in order from smallest to largest: 0.8, 0.6, 0.55, 0.5, 0.555
  3. Put these numbers in order from smallest to largest: 0.9, 0.4, 0.8, 0.11, 0.1
  4. Put these numbers in order from smallest to largest: 0.13, 0.103, 0.301, 0.013
  5. Put these numbers in order from smallest to largest: 0.4444, 0.44444, 0.444, 0.44, 0.4

Ordering Percentages

Percentages are also easy to sort as they’re just values that you can put in numerical order, eg 35% is bigger than 17% because 35 is bigger than 17.

Quiz

  1. Put these numbers in order from largest to smallest: 25%, 12%, 80%, 100%, 4%
  2. Put these numbers in order from largest to smallest: 13%, 103%, 31%, 30%, 30.1%
  3. Put these numbers in order from largest to smallest: 2%, 222%, 22%, 2.2%, 2.22%
  4. Put these numbers in order from largest to smallest: 24%, 4%, 4.4%, 80%, 42%
  5. Put these numbers in order from largest to smallest: 14%, 71%, 3.5%, 5.3%, 4%

Ordering a Mixture

This is where it gets tricky. There’s no single way of comparing fractions, decimals and percentages, so once you’ve numbered the values that are ‘obviously’ bigger and smaller, you’ll have to convert the others into the most common form, eg if there are three fractions, two decimals and a percentage, turn them all into fractions.

This usually saves time, but look out for ‘awkward’ numbers that you can’t easily turn into a different format, eg 0.618 is impossible to turn into a common fraction, and the number π is an ‘irrational number’ that can’t be converted into anything else!

Quiz

  1. Put these numbers in order from smallest to largest: 0.2, 11%, 25%, 1/4, 3/8
  2. Put these numbers in order from smallest to largest: 99.9%, 0.9, 7/8, 8/9, 0.99
  3. Put these numbers in order from smallest to largest: 0.8, 4/5, 5/6, 81%, 90%
  4. Put these numbers in order from smallest to largest: 0.5, 55%, 4/5, 7/8, 77%
  5. Put these numbers in order from smallest to largest: 77%, 0.7, 3/4, 2/3, π