Part a) Divide 210 into the ratio two to five. Part b) Write the ratio nine to 15 in the form one to 𝑛.
In part a of the question then, we’ve been given this amount 210 and asked to divide it into the ratio two to five. This means that we want to divide 210 up into equal parts and then separate them into a group of two equal parts and a group of five equal parts. That will look something like this.
The question is how big are each of these parts. Well, to work this out, we’ll first see how many parts there are in total, the sum of two and five which is seven. Remember that these are equal parts and we’re sharing the full amount of 210 in this ratio. So we know that seven parts are worth 210. We then need to work out what one part is which we do by scaling our ratio down. To get from seven to one, we need to divide by seven.
So to work out the value of one part, we need to divide 210 by seven. We can do this using a calculator or we recall that 21 divided by seven is three. So 210 divided by seven must be 30. So now, we know that one part in our ratio or one little circle on our diagram is worth 30. But we need to know what two parts and five parts are worth.
To get two parts from one part, we can either multiply by two or we can just add one part and one part. In either case, it gives 60. So two parts of our ratio are worth 60. To work out the value of five parts from one part, we need to multiply by five. Three multiplied by five is 15. So 30 multiplied by five is 150. This tells us then that two parts of our ratio are worth 60 and five parts are worth 150.
It’s always sensible to check our work by adding these two values together and confirming that they do indeed give the total amount. 60 plus 150 is indeed equal to 210. We can, therefore, give our answer as the unsimplified ratio 60 to 150. You’ll see that this ratio does in fact cancel down to the ratio of two to five if we divide both sides by 30.
In part b of the question, we’re asked to take the ratio nine to 15 and write it in the form one to 𝑛. So we’re looking for an equivalent ratio but where the left-hand side has been replaced by the number one and the right-hand side has been replaced by a number which we need to work out.
We see that in order to go from nine to one, we must have divided the left-hand side of this ratio by nine. So in order to keep the ratio equivalent, we must also do the same to the right side of the ratio. The number 𝑛 then will, therefore, be the value of 15 divided by nine; that’s the fraction 15 over nine.
We can simplify this fraction there because both 15 and nine are multiples of three. 15 divided by three is five and nine divided by three is three. So 15 over nine simplifies to five over three.
So we find that the ratio nine to 15 can be written in the form one to 𝑛, where 𝑛 is equal to five over three. The ratio nine to 15 is equivalent to the ratio one to five over three.