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Video: Finding the Percentage Change

Tim Burnham

Learn how to calculate the percentage by which a value has increased or decreased. For example, find the percentage discount if a train ticket is reduced from $39 to $30. We emphasize the importance of being clear about the percentage of WHAT?

10:03

Video Transcript

In this video we’re gonna look at some finding the percentage change questions. We’ll talk about what we mean when we say a percentage change. And we’ll also discuss some of the common mistakes that you need to avoid.

We use percentages as a way of representing proportions. They help us to make comparisons between different situations. For example, if we are looking for a new lawn mower and one brand claims that someone save ten minutes cutting their lawn with that mower, while another brand claims that someone save twenty minutes cutting their lawn with that mower, which one should we go for if we want to save time cutting our lawn? Well, it really depends how long the two people would normally spend cutting their lawns in the first place, assuming they both started with the same mower. In all these questions you need to ask yourself compared to what or from what?

Let’s say that the two people were using identical lawn mowers before trying the new brands. And the first one normally took twenty minutes to cut their lawn, while the second one normally took fifty minutes to cut their lawn. So working at the percentage change for the first person, they got a decrease of ten minutes compared to an original time of twenty minutes. So as a proportion that means minus ten over twenty. So that’s minus a half. So they’re decreasing the time that they spend cutting the lawn by a half. And to convert that proportion into percentage, we need to multiply it by a hundred. And that gives us minus fifty percent. So the first person has decreased the amount of time they spend cutting the lawn by fifty percent. And for the second person they’ve saved twenty minutes compared to the original fifty minutes. So the proportion change is minus twenty over fifty. And when we times that by a hundred to convert the proportion to a percentage, that turns out to be a decrease of forty percent.

So although the second person save twenty minutes cutting their lawn compared to only ten minutes that the first person saved, that was a smaller proportion of the overall time that they spent cutting the lawn than the original person. So that means that brand A saves a larger proportion of time when cutting the lawn. And if person B had been using brand A’s new mower, they would have saved fifty percent of their time when cutting their lawn, which means that they’d save twenty-five minutes instead of twenty minutes.

Right then, let’s do some percentage change questions. A train ticket costs thirty-nine dollars if you buy it on the day that you travel. If you buy it before that day, then it’ll only cost thirty dollars. Find the percentage discount for buying early to the nearest percent.

So we’re looking for the percentage discount for buying early. So the original price is thirty-nine dollars. And if we get the discount price, we’re only paying thirty dollars. So I’m going from thirty-nine dollars down to thirty dollars, that’s a saving of nine dollars. So the proportion of the full price that we saved is minus nine over thirty-nine. And if we want to scale that proportion up to a percentage, we need to multiply by a hundred. And when we tap that into our calculator, we get minus twenty-three point O seven six nine two three and so on and so on and so on. So to the nearest whole percentage, that’s twenty-three percent saving. Now the fact that it’s a negative number means that the cost reduced. It means we got a saving or a discount. So my answer is we had a twenty-three percent discount to the nearest one percent.

Now we gotta find the percentage change from four-hundred and ninety-three to three-hundred and eighty-six to the nearest one percent and state whether it’s an increase or a decrease. Well we’re going from four-hundred and ninety-three to three-hundred and eighty-six. And that is a drop of one hundred and seven. So to work out the proportion of the change, well minus one hundred and seven is the change, we started off at four-hundred and ninety-three. So we’re looking at minus one hundred and seven over four-hundred and ninety-three.

And to convert that proportion into a percentage, we just multiply by a hundred. And when we put that into our calculator, we get negative twenty-one point seven O three eight five and then lots of other digits. So ignoring everything after the decimal place, we see that would be minus twenty-one percent. But a sneaky peek at the first decimal place tells us it’s a seven. Now a seven is five or above, so that twenty-one is going to have to round up to a twenty-two. So to the nearest one percent, it’s twenty-two percent. And the fact that it was negative means that the-the number went down — means it’s a decrease. So the answer is twenty-two percent decrease.

Fred paid fifty dollars for a new jumper. He then found that his friend Ted had bought exactly the same type of jumper, but he paid sixty-five dollars for it. Ted said he paid thirty percent more than Fred, but Fred said he paid only twenty-three percent less than Ted. Who was right?

First let’s look at how much more Ted paid than Fred paid. So Fred paid fifty dollars; Ted paid sixty-five dollars. That means compared to Fred, Ted paid fifteen dollars more. So when Ted is comparing himself to Fred, the starting point is fifty dollars. So we paid fifteen dollars more than fifty dollars. And to convert that proportion to a percentage, we need to multiply by a hundred. So that’s an increase of thirty percent.

And Ted said that he paid thirty percent more than Fred. Yeah well that is correct. But when Fred said that he paid twenty-three percent less than Ted, he was comparing himself with a starting point of Ted’s price, which was sixty-five dollars. So the difference in price is still fifteen dollars. Fred paid fifteen dollars less than Ted.

But when we’re working out the proportion, that fifteen dollars is being compared to the sixty-five dollars starting point of what Ted paid for the jumper. And again we multiply it by a hundred to convert that to a percentage. And that is minus twenty-three point O seven six nine two three and then lots of other digits percent. Well to the nearest one percent, Fred did in fact pay twenty-three percent less than Ted. So it looks like he was right too. So to answer the question “who was right?” they were both right.

In this question, according to the United Nations estimates the world’s population was as shown in the table below in nineteen hundred, nineteen fifty, and the year two thousand. We need to calculate the percentage increases between nineteen hundred and nineteen fifty and between nineteen fifty and two thousand. And we need to get our answers to one decimal place. So the figures we’ve got: in nineteen hundred, it was one point six five billion, in nineteen fifty, it was two point five two billion, and by two thousand, it was six point O six billion.

Well we’re gonna do two different calculations then: going from nineteen hundred to nineteen fifty and then from nineteen fifty to two thousand. Well between nineteen hundred and nineteen fifty, the population went from one point six five billion up to two point five two billion, which was an increase of nought point eight seven billion. And between nineteen fifty and two thousand, it went from two point five two billion to six point O six billion, which is an increase of three point five four billion.

And for the first calculation, the increase was positive nought point eight seven billion. Now that was compared to the starting point of one point six five billion. So that’s the proportion. And to convert it into a percentage, we’re gonna multiply by a hundred. And that gives us an increase of positive fifty-two point seven two seven two lots of other digits percent.

Then between nineteen fifty and two thousand, the increase was three point five four billion, but that was compared to a starting point of two point five two billion. Then multiplying by a hundred to convert that to a percentage, we get positive a hundred and forty point four seven six one nine zero then lots of other digits percent.

Well looking back at the question, we’ve got to give our answers correct to one decimal place. And looking at the first answer there, fifty-two point seven two, so fifty-two point seven to one decimal place. The next digit is a two, so we don’t have to round that up. And in the second case there between nineteen fifty and two thousand, a hundred and forty point four. But the next digit is a seven, so that four is gonna have to round up to a five. So there’re our answers: nineteen hundred to nineteen fifty was an increase of fifty-two point seven percent and from nineteen fifty to two thousand, that’s an increase of a hundred and forty point five percent.

Now a little bit of interpretation on that, between nineteen hundred and nineteen fifty the population went up by just over fifty percent. So half as many people who were living in nineteen hundred again were added to the population by nineteen fifty. But between nineteen fifty and two thousand, the- the population increased by more than a hundred percent. So the population in that fifty years more than doubled; in fact it went up by nearly one and a half times in that time.

Now we saw in our first example that having a bigger number here — a bigger raw increase or decrease — didn’t necessarily mean that there was a bigger proportional change you have to compare it to the starting point. So in the lawn mower case even though we had a bigger saving of twenty minutes, that was a smaller proportion of the starting point. In this case a bigger increase of three point five four billion was more, even compared to the starting point here. It was a hundred and forty percent rather than only fifty-two point seven percent.

So you do have to complete your percentage calculations to get the full picture and make a better comparison between those two sets of figures.