Tag Archives: fractions

Working out values from a pie chart

Working out Values from a Pie Chart

This is a typical question from a Dulwich College 11+ Maths paper that asks you to work out various quantities from a pie chart.

To answer questions like this, you have to be comfortable working with fractions and know that there are 360 degrees in a circle.

So how should you start?

The first question asks for the fraction of the school children who liked tennis.

To work this out, you just need to take the following steps:

  1. Put the number of degrees showing the tennis segment over 360 to create a fraction.
  2. Simplify the fraction.

The number of degrees is 45, so the fraction is 45/360.

The first step to simplifying fractions is to see if the numerator goes into the denominator, which it does in this case: 45/45 = 1 and 360/45 = 8, so the fraction is 1/8 in its lowest terms.

(By the way, for a complete guide to simplifying fractions, just read Working with Fractions.)

The second question asks how many of the children preferred cricket.

To answer this, you should be able to learn a bit from the first question.

To work this out, you just need to take the following steps:

  1. Put the number of degrees showing the cricket segment over 360 to create a fraction.
  2. Multiply that fraction by the number of school children in the survey, which is 240.

As with the first question, you need to work out the fraction of the children in the survey you’re dealing with.

In this case, it’s 60/360 or 1/6.

To find out the number of children, you just have to multiply by 240, which is 1/6 x 240 = 40.

The final question asks you to estimate (or guess) how many of the children would say their favourite sport was football out of the whole school of 1200 pupils.

To work this out, you just need to take the following steps:

  1. Work out the number of degrees taken up by the football segment of the pie chart.
  2. Put the number of degrees over 360 to create a fraction.
  3. Multiply that fraction by the number of children in the school, which is 1200.

To work out the number of degrees, it’s easier if you spot that the first half of the pie chart is composed of just football and tennis.

There are 180 degrees in total for that half, so taking away 45 degrees for the tennis-lovers gives you 135 degrees.

This works out at a fraction of 135/360 or 3/8.

Now, we only have data for the 240 children who’ve been surveyed, but that’s why we’re being asked to estimate the answer.

We have to assume that the other kids at school share the same preferences as the ones in the survey.

If we do that, all we need to do is multiply 3/8 by 1200 to get 3/8 x 1200 = 450.

And that’s it…!

 

 

 

 

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Fractions, decimals and percentages

Working with Fractions

People don’t like fractions. I don’t know why. They’re difficult to begin with, I know, but a few simple rules will help you add, subtract, multiply and divide.

Adding and Subtracting Fractions

Adding and subtracting are usually the easiest sums, but not when it comes to fractions. If fractions have the same denominator (the number on the bottom), then you can simply add or subtract the second numerator from the first, eg 4/5 – 3/5 = 1/5. If not, it would be like adding apples and oranges.

They’re just not the same, so you first have to convert them into ‘pieces of fruit’ – or a common unit. The easiest way of doing that is by multiplying the denominators together. That guarantees that the new denominator is a multiple of both the others.

Once you’ve found the right denominator, you can multiply each numerator by the denominator from the other fraction (because whatever you do to the bottom of the fraction you have to do to the top), add or subtract them and then simplify and/or convert into a mixed number if necessary, eg 2/3 + 4/5 = (2 x 5 + 4 x 3) / (3 x 5) = (10 + 12) / 15 = 22/15 = 1 7/15.

  1. Multiply the denominators together and write the answer down as the new denominator
  2. Multiply the numerator of the first fraction by the denominator of the second and write the answer above the new denominator
  3. Multiply the numerator of the second fraction by the denominator of the first and write the answer above the new denominator (after a plus or minus sign)
  4. Add or subtract the numerators and write the answer over the new denominator
  5. Simplify and/or turn into a mixed number if necessary

Note that you can often use a simpler method. If one of the denominators is a factor of the other, you can simply multiply the numerator and denominator of that fraction by 2, say, so that you get matching denominators, eg 1/5 + 7/10 = 2/10 + 7/10 = 9/10. This means fewer steps in the calculation and lower numbers, and that probably means less chance of getting it wrong.

Sample Questions

  1. 1/5 + 2/3
  2. 3/8 + 11/12
  3. 13/24 – 5/12
  4. 7/8 – 3/4
  5. 5/8 – 2/3

Multiplication

This is the easiest thing to do with fractions. You simply have to multiply the numerators together, multiply the denominators together and then put one over the other, simplifying and/or converting into a mixed number if necessary, eg 2/3 x 4/5 = (2 x 4) / (3 x 5) = 8/15.

  1. Multiply the numerators together
  2. Multiply the denominators together
  3. Put the result of Step 1 over the result of Step 2 in a fraction
  4. Simplify and/or turn into a mixed number if necessary

Sample questions

  1. 1/5 x 2/3
  2. 7/12 x 3/8
  3. 4/5 x 2/3
  4. 4/9 x 3/4
  5. 5/8 x 2/3

Division

Dividing by a fraction must have seemed like a nightmare to early mathematicians, because nobody ever does it! That’s right. Nobody divides by a fraction, because it’s so much easier to multiply.

That’s because dividing by a fraction is the same as multiplying by the same fraction once it’s turned upside down, eg 2/3 ÷ 4/5 = 2/3 x 5/4 = (2 x 5) / (3 x 4) = 10/12 = 5/6. You can even cut out the middle step and simply multiply each numerator by the denominator from the other fraction, eg 2/3 ÷ 4/5 = (2 x 5) / (3 x 4) = 10/12 = 5/6.

  1. Multiply the numerator of the first fraction by the denominator of the second
  2. Multiply the numerator of the second fraction by the denominator of the first
  3. Put the result of Step 1 over the result of Step 2 in a fraction
  4. Simplify and/or turn into a mixed number if necessary

Note: In some cases, it’s possible to divide one numerator by the other and one denominator by the other, but the answers must both be whole numbers for it to work, eg 8/9 ÷ 2/3 = (8 ÷ 2) / (9 ÷ 3) = 4/3 or 1 1/3, but 7/9 ÷ 4/5 doesn’t work because 4 doesn’t go into 7 evenly and 5 doesn’t go into 9 evenly.

The advantage of the second method is that you don’t have to spend so much time simplifying the resulting fraction as the numbers are divided rather than multiplied, which makes them smaller. If we cross-multiplied using the standard method, we’d end up with 24/18 rather than 4/3. However, the benefit of the standard method is that it always works!

Sample Questions

  1. 1/5 ÷ 2/3
  2. 2/7 ÷ 3/5
  3. 4/7 ÷ 2/3
  4. 7/8 ÷ 3/4
  5. 5/6 ÷ 2/3

Simplifying Fractions

One way of simplifying fractions is to divide by the lowest possible prime number over and over again, but that takes forever! It’s much simpler to divide by the Highest Common Factor (or HCF), which is either the numerator itself or half of it or a third of it etc:

  1. If possible, divide both the numerator and the denominator by the numerator. If that works, you’ll end up with a ‘unit fraction’ (in other words, 1 over something) that can’t be simplified any more, eg 7/14 = 1/2 because 7 ÷ 7 = 1 and 14 ÷ 7 = 2.
  2. If the numerator doesn’t go into the denominator, try the smallest fraction of the numerator (usually a half or a third) and then try to divide the denominator by the result, eg 24/36 = 2/3 because half of 24 is 12, and 36 ÷ 12 = 3.
  3. If that doesn’t work, keep repeating Step 2 until you find the answer, eg 24/30 = 4/5 because a quarter of 24 is 6, and 30 ÷ 6 = 5 (and a half and a third of 24 don’t go into 30).

Sample Questions

  1. Simplify 14/28
  2. Simplify 8/24
  3. Simplify 30/50
  4. Simplify 27/36
  5. Simplify 45/72

Turning Improper Fractions into Mixed Numbers

To turn an improper fraction into a mixed number, simply divide the numerator by the denominator to find the whole number and then put the remainder over the original denominator and simplify if necessary, eg 9/6 = 1 3/6 = 1 1/2.

  1. Divide the numerator by the denominator
  2. Write down the answer to Step 1 as a whole number
  3. Put any remainder into a new fraction as the numerator, using the original denominator
  4. Simplify the fraction if necessary

Sample Questions

  1. What is 22/7 as a mixed number?
  2. What is 16/5 as a mixed number?
  3. What is 8/3 as a mixed number?
  4. What is 18/8 as a mixed number?
  5. What is 13/6 as a mixed number?

Turning Mixed Numbers into Improper Fractions

To turn a mixed number into an improper fraction, multiply the whole number by the denominator of the fraction and add the existing numerator to get the new numerator while keeping the same denominator, eg 2 2/5 = (10 + 2)/5 = 12/5.

  1. Multiply the whole number by the denominator of the fraction
  2. Add the answer to the existing numerator to get the new numerator
  3. Write the answer over the original numerator
  4. Simplify if necessary

Sample Questions

  1. What is 2 2/7 as an improper fraction?
  2. What is 3 2/3 as an improper fraction?
  3. What is 4 1/4 as an improper fraction?
  4. What is 5 1/5 as an improper fraction?
  5. What is 3 2/9 as an improper fraction?

There you go. Easy peasy lemon squeezy!

 

 

 

 

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Fractions, decimals and percentages

Fractions, Decimals and Percentages

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 

     

     

     

    If you’re looking for past papers with answers, especially in the run-up to 11+/13+ exams, GCSEs or A-levels, you can visit my Past Papers page and subscribe for just £37.99 a year.