Tip:
Highlight text to annotate it
X
We can use ratios to solve certain kinds of problems involving percentages. In particular,
questions where we have to calculate the total from a part, or one size part from another
part. To answer these kinds of questions, think carefully through: What percentage have
I got now? What percentage do I want? And remember, the original amount is always one
hundred percent.
Let’s start with total from part. In a “twenty percent off” sale, a pair of shoes was sold
for forty-five dollars. What was the price before the twenty percent discount was applied?
Now, the original price was a hundred percent. But do I have the original price? No, I’ve
got the price the shoes were sold for, after the twenty percent discount was applied. So
what percentage have I got? I’ve got eighty percent, the original hundred percent minus
the twenty percent discount. Okay, so let’s set up a ratio to solve this. Eighty percent
of the original price was forty-five dollars. What percentage do I want? I want the original
price, which is a hundred percent. So, how do I convert eighty into a hundred? Well,
multiply by the new number over the old number: a hundred divided by eighty. So, multiply
the other side, the price, by the same fraction: a hundred over eighty. You’ll want a calculator
for this. The original price of the shoes was fifty-six dollars and twenty-five cents.
Now I know some of you are thinking, why did we have to do all that? Couldn’t we just
get twenty percent of the forty-five dollars, and add that on to get the original price?
Well, why don’t we try that and see what happens? Twenty percent of forty-five is nine.
Forty-five plus nine is fifty-four. Hmm, that’s not the same as the fifty-six twenty-five
we got before, is it? Why not? Why didn’t that work?
I’m going to let you keep thinking about that. See if you can work it out. Just to
prove to you that my first answer is the correct one, here’s what happens if we take twenty
percent of it, and then subtract that as the discount: we get forty-five dollars, which
is correct. So why didn’t that work the other way?
The bottom line is this: Always ask yourself, do you have the original amount? If not, what
percentage have you got?
Let’s try another example, total from part. A retailer sells a fridge for eight hundred
and forty dollars and makes a sixty percent profit. What did the retailer pay for the
fridge? The original price was a hundred percent. Do I have the original price?
No, I have a later price: the price the retailer sold it for. The original price is the amount
the retailer paid for it, which is what I have to find out. So I can’t just get sixty
percent of eight hundred and forty, and take that off. Eight hundred and forty is not the
original amount.
What percentage have I got? Well, the original was a hundred percent, plus the sixty percent
profit makes a hundred and sixty percent. Eight hundred and forty dollars is one hundred
and sixty percent of the original price. I need to find a hundred percent, the original
amount. Okay, so I need to multiply by a hundred over a hundred and sixty. Eight hundred and
forty times a hundred over a hundred and sixty is (with a calculator) five hundred and twenty-five
dollars. The retailer paid five hundred and twenty-five dollars for the fridge.
Can we check that answer? Well, what’s sixty percent of five hundred and twenty-five? It’s
three hundred and fifteen. And if we add those two together, we get back to eight hundred
and forty dollars. Good.
We can also use ratios to solve percentage problems where I want to find out something
other than the original amount. We call this a “part from part” problem, and they often
come about from questions involving GST. Here’s an example. A formal dress is sold for two
hundred and fifty-five dollars, including ten percent GST. (That’s the standard tax
rate in Australia.) The retailer has to send the money for the GST to the tax office. How
much GST was included in the price of this dress?
Well, the original price was a hundred percent. Do I have the original price? No. That would
be the price before the GST was added. I have the price including the GST. That’s the
original one hundred percent, plus ten percent GST, making a hundred and ten percent. Two
hundred and fifty-five dollars is one hundred and ten percent. Okay, now what percent do
I want? I want the amount of the GST, which is ten percent. How will I get from a hundred
and ten percent to ten percent? Well, multiply by the new figure over the old: ten divided
by a hundred and ten. So, multiply the price by ten over a hundred and ten as well. And
if you get your calculator and do that, the amount you get is twenty-three dollars and
eighteen cents (rounded off to the nearest cent). The GST that was included in the price
of that dress was twenty-three dollars and eighteen cents.
See how the same method, using ratios, can be used to go from one percentage to another,
even without first calculating the original hundred percent amount?
Let’s do another example of this, part from part. If you visit England and buy a souvenir,
you will be charged seventeen and a half percent VAT (or value-added-tax, which is the English
equivalent of our GST). You can claim the VAT back if you take the object out of the
country when you leave. Suppose you buy a toy London Bus for seven pounds fifty. How
much can you get back?
Well, the original price was a hundred percent. But do I have the original price? No, I’ve
got the price I paid for it, which includes the VAT. What percentage is that? It’s a
hundred percent plus the seventeen and a half percent VAT, which makes it a hundred and
seventeen point five percent of the original price. Okay, what percentage do I want? I
need to work out how much I can get back when I leave the country, which is the amount of
the VAT, or seventeen and a half percent.
So, how do I go from a hundred and seventeen point five percent, to seventeen point five
percent? Multiply by the new number over the old. So, do the same to the price on the other
side: seven point five, times seventeen point five over a hundred and seventeen point five,
gives me one point one two, one pound twelve (rounded off to two decimal places as usual).
Be careful with your calculator, remember, make sure you do this in one calculation;
don’t round off that fraction, or you’ll get the wrong answer. The VAT I can get back
on my souvenir is one pound twelve p.
And that’s how to use ratios to solve total-from-part and part-from-part percentage problems.