Eva is practising a one and two-fifths of a minute gymnastics floor routine. The dance elements of her routine last three-quarters of a minute. What fraction of her routines do the dance elements represent? Give your answer in its simplest form.
There are two different methods we can try here. Both methods rely on the fact that to find the fraction of the routine that the dance elements represent, we divide the time spent on the dance elements by the total length of the routine.
For the first method, we do this by considering the amount of time each part of her routine takes in seconds. Remember one minute is equal to 60 seconds. We can begin by working out what two-fifths of a minute is. Once we have that, we’ll be able to work out the total length of the floor routine in seconds.
To find two-fifths of a minute in seconds, we need to find two-fifths of 60 seconds. To do that, we begin by calculating one-fifth of 60.
To find one-fifth of 60, we need to split 60 into five equal parts. We divide 60 by five which is 12. And that tells us that one-fifth of a minute or one-fifth of 60 seconds is 12 seconds.
We can then find the value of two-fifths by multiplying the value of one-fifth by two. 12 multiplied by two is 24. So two-fifths of 60 is 24. And this means that two-fifths of a minute is equal to 24 seconds.
The whole floor routine takes one and two-fifths of a minute; that’s 60 seconds for the one whole minute and 24 seconds for the two-fifths. Altogether, the floor routine takes 84 seconds.
Eva spends three-quarters of a minute performing the dance elements. To find three-quarters of a minute in seconds, we need to find three-quarters of 60.
This time, we work out the value of one-quarter of 60 by dividing it by four. And another way of dividing by four is dividing by two and then dividing by two again. 60 divided by two is 30 and 30 divided by two is 15.
This means that one-quarter of 60 is 15. And a quarter of a minute is 15 seconds. To find the number of seconds in three-quarters of a minute then, we need to multiply 15 by three.
Now, you might be able to work out 15 multiplied by three in your head. But let’s look at what happens if we use the column method.
Five multiplied by three is 15. So we put a five in the units column and carry the one. One multiplied by three is three. And then when we add the one, we get four. That tells us that three-quarters of 60 is 45. And three-quarters of a minute is equal to 45 seconds.
We said to find a fraction of the routine that the dance elements represent, we divide the time spent on dance elements by the time spent on the total routine. That’s 45 divided by 84.
We have been told though to give our answer in its simplest form. Currently, 45 and 84 share a common factor. They can both be divided by three.
A good way to check for divisibility of three is to add the digits of the number. And if that number can be divided by three, the original number can be divided by three. The digits of 84, for example, are eight and four. And when we add eight and four, we get 12, which we know it’s divisible by three. So that means that 84 is divisible by three.
We said earlier that three multiplied by 15 was 45. So that means that 45 divided by three is 15. And then, we can use the bus stop method to divide 84 by three. Eight divided by three is two with a remainder two. And 24 divided by three is eight.
So 84 divided by three is 28. And in its simplest form, the fraction of Eva’s routine that the dance elements represent is 15 over 28.
Let’s now consider method two. Remember we said that to find the fraction of the routine that the dance elements represent, we divide the time spent on the dance elements by the time for the total routine. That’s three-quarters divided by one and two-fifths.
When performing fraction arithmetic, we need to make sure that any mixed numbers are converted into improper fractions.
To do that, we begin by multiplying the integer part of our number by the denominator. That’s one multiplied by five. We then add the numerator which gives us seven. The denominator always stays the same. So one and two-fifths is equivalent to seven-fifths. And that means we can write our division as three-quarters divided by seven-fifths.
To divide by a fraction, we keep the first number the same. We change the division to a multiplication and we find the reciprocal of the final fraction. That’s sometimes called flipping fraction. And you might have seen the letters KCF as a way of remembering this process.
So in this case, three-quarters stays as it is. The division changes to a multiplication. And seven-fifths becomes five over seven. We then multiply as normal. Three multiplied by five is 15 and four multiplied by seven is 28.
Once again, we’ve shown that the fraction of the routine that the dance elements represent in its simplest form is 15 over 28.