Tag Archives: fractions

Working with fractions

Me again...

Me again…

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

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

Sample questions

  1. 1/5 + 2/3
  2. 2/7 + 3/5
  3. 4/5 – 2/3
  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. 2/7 x 3/5
  3. 4/5 x 2/3
  4. 7/8 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

Sample questions

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

Simplifying fractions

By the way, to simplify a fraction, try dividing the denominator by the numerator first, eg 9/18 = 1/2. If that works, you don’t have to do anything else. If not, try dividing by the first few prime numbers, ie 2, 3, 5, 7 and 11. You don’t need to try the other numbers, because they’re all multiples of the primes, so they won’t work if the others don’t, eg 4 won’t work if 2 doesn’t work. Ideally, the quickest way would be to divide the numerator and denominator by the highest common factor (or HCF), but you don’t know what that is at the beginning, so it would take time to work it out. This way is a good compromise.

  1. If possible, divide the numerator and denominator by the numerator
  2. If the numerator doesn’t go exactly, start dividing by the smallest prime number that will go into both numbers, starting with 2, 3, 5, 7 and 11
  3. Repeat Step 2 until the only number that goes into the numerator and denominator is 1

Sample questions

  1. Simplify 14/28
  2. Simplify 8/24
  3. Simplify 4/12
  4. Simplify 27/36
  5. Simplify 30/50

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

“Now, who wants 5/6 of this…?”

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 short cut.

  • 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.

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%.

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.

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%.

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.

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.