// Numbas version: exam_results_page_options {"name": "Numerical fractions", "metadata": {"description": "

Equivalent fractions, simplifying, adding, subtracting, multiplying and dividing fractions. Converting between mixed numbers and improper fractions.

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{Name[0][0]} has written $\\frac{\\var{num5}}{\\var{denom5}}$ in the equivalent form $\\frac{\\var{num6}}{\\var{denom6}}$.

What has {Name[0][0]} done to the first fraction in order to get the second? {Name[0][1]} has multiplied the top and bottom by [[0]] .

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If we multiply the top and bottom of a fraction by a number (not zero) we get an equivalent fraction. We say equivalent because they represent the same amount of the whole.

For example, suppose you cut a cake up into 3 parts and throw away one piece, what is left is two thirds of the whole cake, that is, $\\frac{2}{3}$ of the whole cake. Now suppose you have another identical cake, this time you cut it into 6 parts and throw away two parts, what is left is four sixths of the whole cake, that is, $\\frac{4}{6}$ of the whole cake.

Notice in both situations you end up with the same amount of cake!

So $\\frac{2}{3}$ is equivalent to $\\frac{4}{6}$ and we can write \\[\\frac{2}{3}=\\frac{4}{6}\\]

If you look at the numbers you might notice that for the second cake we just doubled all the numbers, and in the second fraction all the numbers are two times those in the first fraction. In general equivalent fractions are formed by multiplying (or dividing) the top and bottom of a fraction by the same number.

So if you were asked how a person got from $\\frac{5}{6}$ to the equivalent fraction $\\frac{20}{24}$ you ask yourself 'what do I multiply 5 by to get 20?' and 'what do I multiply 6 by to get 24?' and then realise they must have done the following

\\[\\frac{5}{6}=\\frac{5\\times 4}{6\\times 4}=\\frac{20}{24}\\]

{Name[1][0]} has written $\\frac{\\var{num2*mult5}}{\\var{denom2*mult5}}$ in the equivalent form $\\frac{\\var{num2}}{\\var{denom2}}$.

What has {Name[1][0]} done to the first fraction in order to get the second? {Name[1][1]} has divided the top and bottom by [[0]] .

If we divide the top and bottom of a fraction by a number (not zero) we get an equivalent fraction. We say equivalent because they represent the same amount of the whole.

For example, suppose you cut it into 6 parts and throw away two parts, what is left is four sixths of the whole cake, that is, $\\frac{4}{6}$ of the whole cake. Now suppose you have another identical cake, this time you cut a cake up into 3 parts and throw away one piece, what is left is two thirds of the whole cake, that is, $\\frac{2}{3}$ of the whole cake.

Notice in both situations you end up with the same amount of cake!

So $\\frac{4}{6}$ is equivalent to $\\frac{2}{3}$ and we can write \\[\\frac{4}{6}=\\frac{2}{3}\\]

If you look at the numbers you might notice that for the second cake we just halved all the numbers, and in the second fraction all the numbers are half of those in the first fraction. In general equivalent fractions are formed by dividing (or multiplying) the top and bottom of a fraction by the same number.

So if you were asked how a person got from $\\frac{20}{24}$ to the equivalent fraction $\\frac{5}{6}$ you ask yourself 'what do I divide 20 by to get 5?' and 'what do I divide 24 by to get 6?' and then realise they must have done the following

\\[\\frac{20}{24}=\\frac{20\\div 4}{24\\div 4}=\\frac{5}{6}\\]

Please enter numbers to create equivalent fractions.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

{num1}	=	[[0]]	=	{num3}	=	[[2]]
{denom1}		{denom2}		[[1]]		{denom4}

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These are equivalent fractions so the same number that multiplied the numerator must multiply the denominator.

For example given:

\\[\\frac{8}{5}=\\frac{}{15}\\]

you can see the denominator of 5 was multiplied by 3 to become 15, so to make an equivalent fraction, we would need to multiply the numerator by 3 as well. So the blank must be $8\\times 3$ which is $24$. Your working might look like this:

\\[\\frac{8}{5}=\\frac{8\\times 3}{5\\times 3}=\\frac{24}{15}\\]

Simplify the following fractions into their lowest forms:

(for this question, if your fraction turns into a whole number write it over 1)

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

{num2*mult6}	=	[[0]]
{denom4*mult6}		[[1]]

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n

{num3*mult3}	=	[[2]]
{denom1*mult3}		[[3]]

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Divide the top and bottom by their highest common factor, or repeatedly divide the top and bottom by common factors.

We can write a fraction in a lower form by dividing the top and bottom by the same number. We can repeatedly do this until there are no numbers that 'go evenly into' both the top and the bottom. At this point, the number is in 'lowest form'.

For example:

To simplify $\\frac{360}{132}$ we might first notice that we can divide both the top and bottom by 2 to get $\\frac{180}{66}$, then you might realise you can divide both the top and bottom by 6 to get $\\frac{30}{11}$. At this point, there is no number that will divide both evenly because they have no common factor other than 1. Your working could look like this:

\\[\\frac{360}{132}=\\frac{360\\div 2}{132\\div 2}=\\frac{180}{66}=\\frac{180\\div 6}{66\\div 6}=\\frac{30}{11}\\]

Notice dividing by 2 and then by 6 is the same as dividing by 12, in this example 12 is the highest common factor of the top and the bottom of the fraction.

In general, if you can determine the highest common factor of the two numbers in the fraction you should then divide the top and bottom of the fraction by this number, then there will be no common factors left to divide by and the fraction will be in its lowest form.

For example:

To simplify $\\frac{360}{132}$ you determine the highest common factor of $360$ and $132$ is $12$, and so divide the top and bottom of the fraction by $12$ to get $\\frac{30}{11}$. Your working could look like this:

\\[\\frac{360}{132}=\\frac{360\\div 12}{132\\div 12}=\\frac{30}{11}\\]

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Learn from your mistakes and have another attempt by clicking on 'Try another question like this one' until you get full marks.

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$\\displaystyle\\frac{\\var{a}}{\\var{b}}+\\frac{\\var{c}}{\\var{b}}=$[[0]]

$\\displaystyle\\frac{\\var{d}}{\\var{c}}-\\frac{\\var{a}}{\\var{c}}=$[[1]]

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Add the tops, leave the bottom the same.

These fractions have a common denominator (the number on the bottom). This means they are out of the same number of parts and can be compared easily, for example, it is clear $\\frac{2}{3}$ is less than $\\frac{5}{3}$ but not so clear that $\\frac{3}{5}$ is less than $\\frac{2}{3}$.

Let's say you need to evaluate $\\frac{2}{3}+\\frac{5}{3}$, in words this is 'two thirds plus five thirds', so how many thirds are there in total? Seven thirds!

So we have

\\[\\frac{2}{3}+\\frac{5}{3}=\\frac{2+5}{3}=\\frac{7}{3}\\]

The same logic is used for subtraction. Suppose you had seven fourths and someone borrowed three fourths, then you are left with four fourths.

That is

\\[\\frac{7}{4}-\\frac{3}{4}=\\frac{7-3}{4}=\\frac{4}{4}=1\\]

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$\\displaystyle\\simplify{{f}/{g}+{h}/{j}}=$[[0]]

$\\displaystyle\\simplify{{h}/{f}-{j}/{g}}=$[[1]]

$\\displaystyle \\frac{\\var{a}}{\\var{d}}+\\var{f}=$[[2]]

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Rewrite the fractions so they have a common denominator. Then perform the addition or subtraction as required.

If your question was $\\frac{5}{4}+\\frac{3}{8}$ we could rewrite the first fraction as $\\frac{10}{8}$ (by multiplying the top and bottom by 2) and then both fractions would have a denominator of 8. At this point, we can perform the addition. Our working might look like this:

\\[\\frac{5}{4}+\\frac{3}{8}=\\frac{5\\times 2}{4\\times 2}+\\frac{3}{8}=\\frac{10}{8}+\\frac{3}{8}=\\frac{13}{8}\\]

Often we need to rewrite both fractions to get a common denominator, for instance, $\\frac{5}{4}-\\frac{2}{3}$. We could multiply the first fraction by 3 on the top and bottom, so that it's denominator was 12, and then multiply the second fraction by 4 on the top and bottom so that it also had a denominator of 12. Then we could perform the subtraction. Our working might look like this:

\\[\\frac{5}{4}-\\frac{2}{3}=\\frac{5\\times 3}{4\\times 3}-\\frac{2\\times 4}{3\\times 4}=\\frac{15}{12}-\\frac{8}{12}=\\frac{7}{12}\\]

Also, recall that whole numbers are just fractions with a denominator of 1, for example $3=\\frac{3}{1}$.

In general, the best denominator is the lowest common multiple (LCM) of the two denominators.

$\\displaystyle\\frac{\\var{a}}{\\var{b}}\\times \\frac{\\var{c}}{\\var{d}}=$[[0]]

$\\displaystyle -\\frac{\\var{f}}{\\var{j}}\\times \\var{d}=$[[1]]

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Multiply the tops and the bottoms.

For example

\\[\\frac{4}{5}\\times \\frac{2}{3}=\\frac{4\\times 2}{5 \\times 3}=\\frac{8}{15}\\]

Also recall that whole numbers are just fractions with a denominator of 1, for example $7=\\frac{7}{1}$.

$\\displaystyle \\frac{\\var{f}}{\\var{h}}\\div \\frac{\\var{g}}{\\var{j}}=$[[0]]

$\\displaystyle \\frac{\\var{b}}{\\var{c}}\\div \\var{d}=$[[1]]

$\\displaystyle \\var{j}\\div \\left(\\frac{\\var{-d}}{\\var{f}}\\right)=$[[2]]

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Flip the second fraction and then multiply.

Flipping a fraction is also known as taking the reciprocal of the fraction (or inverting a fraction). Note that a whole number is also a fraction with a denominator of 1, for example, $6=\\frac{6}{1}$.

How do you find half of a number? You could 'divide it by 2', or you could 'multiply by $\\frac{1}{2}$. Notice that $\\frac{1}{2}$ is the reciprocal of 2. When we divide by a number this is actually the same as multiplying by its reciprocal.

Suppose you need to evaluate $\\frac{3}{7}\\div\\frac{5}{4}$. Recall this is the same as asking 'how many $\\frac{5}{4}$s are in $\\frac{3}{7}$?', but that doesn't seem to be very helpful here! What is helpful is realising that dividing by $\\frac{5}{4}$ is the same as multiplying by $\\frac{4}{5}$. Our working could look like this

\\[\\frac{3}{7}\\div\\frac{5}{4}=\\frac{3}{7}\\times\\frac{4}{5}=\\frac{3\\times 4}{7\\times 5}=\\frac{12}{35}\\]

$\\displaystyle \\frac{\\frac{\\var{b}}{\\var{c}}}{ \\frac{\\var{a}}{\\var{d}}}=$[[0]]

$\\displaystyle \\frac{\\frac{\\var{d}}{\\var{g}}}{\\var{f}}=$[[1]]

$\\displaystyle \\frac{\\var{j}}{\\frac{\\var{h}}{\\var{c}}}=$[[2]]

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The fraction bar means division.

The fraction $\\frac{2}{3}$ means 2 divided by 3. So these questions are just division questions! It is important to note which fraction bar is big and which are small, so you know the order of the divisions.

Here are some examples:

\\[\\frac{7}{\\frac{5}{6}}=7\\div\\frac{5}{6} =7\\times\\frac{6}{5}=\\frac{42}{5}\\]

\\[\\frac{\\frac{7}{5}}{6}=\\frac{7}{5}\\div 6=\\frac{7}{5}\\times \\frac{1}{6}=\\frac{7}{30}\\]

\\[\\frac{\\frac{9}{11}}{\\frac{5}{3}}=\\frac{9}{11}\\div\\frac{5}{3}=\\frac{9}{11}\\times \\frac{3}{5}=\\frac{27}{55}\\]

Evaluate the following and write your answer as a fraction or whole number (not a decimal). Use / to signify a fraction or division, for example $\\frac{2}{3}$ is written 2/3.

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Add, subtract, multiply and divide numerical fractions.

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Convert the following improper fractions to mixed numerals (also known as mixed numbers):

Note: Write the whole number part in the first box and the fraction part in the second.
For example, if your answer was $2\\frac{3}{4}$, enter $2$ in the first box and $3/4$ in the second.

$\\displaystyle\\frac{\\var{a1*c1+b1}}{\\var{c1}}=$[[0]][[1]]

$\\displaystyle\\frac{\\var{a2*c2+b2}}{\\var{c2}}=$[[2]][[3]]

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Do the division and write your remainder over the original denominator. Simplify the fraction if possible.

For example, converting $\\frac{21}{9}$ into a mixed numeral, you ask yourself \"how many times does 9 go into 21?\", it goes in twice (since $2\\times 9=18$ but $3\\times 9=27$), with a remainder of 3 (since $21-18=3$). So we can write our answer as $2\\frac{3}{9}$ (which actually means $2+\\frac{3}{9}$). But notice we can simplify the fraction, so we should rewrite our answer as $2\\frac{1}{3}$.

Note: we could have cancelled common factors at the beginning.

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Convert the following mixed numerals to improper fractions:

$\\var{a3}\\frac{\\var{b3}}{\\var{c3}}=$[[0]]

$\\var{a4}\\frac{\\var{b4}}{\\var{c4}}=$[[1]]

Multiply the whole number and the denominator, add the numerator, and put it all over the denominator.

For example $2\\frac{3}{4}$ can be written as $\\frac{2\\times 4+3}{4}$ that is, $\\frac{11}{4}$.

To understand why, realise that $2\\frac{3}{4}$ is shorthand for $2+\\frac{3}{4}$ and if we want to add these numbers we need to have a common denominator (recall the denominator of a whole number is 1). Our working could look like this:

\\[2\\tfrac{3}{4}=2+\\frac{3}{4}=\\frac{2\\times 4}{4}+\\frac{3}{4}=\\frac{2\\times 4+3}{4}=\\frac{11}{4}\\]

but in practice we normally don't write anything more than \\[2\\tfrac{3}{4}=\\frac{2\\times 4+3}{4}=\\frac{11}{4}\\]

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