Lesson video

Hi, I'm Mrs. Wheelhouse.

Welcome to today's lesson on finding the original amount after an increase.

This lesson falls within the unit understanding multiplicative relationships, percentages, and proportionality.

Let's get started.

By the end of today's lesson, you'll be able to calculate the original value, given the final value after a stated percentage increase.

Hmm.

What do I mean by that? Well, let's find out in our lesson.

We're gonna be using the keyword reciprocal today, and you can see the definition for reciprocal on the screen now.

Feel free to pause if you want time to reread this before you get going in the lesson.

Today's lesson is broken into two parts, and we're gonna start with the first section on finding the original value.

Alex gives 30% of his sweets to his brother.

Oh, that's quite nice, Alex.

Although if you are Alex's brother, you're probably thinking that 30% does not seem that fair.

His brother gets 15 sweets.

Hmm, actually that does seem like quite a lot.

Well, how many sweets did Alex have to begin with? Hmm? How are we gonna approach this? Well, let's set up a bar model.

What would be a sensible partition for our bar model? Well, 10 parts would seem sensible because we're talking about 30%.

There we go.

I've broken my bar into 10 equal parts.

What will each part represent? Well, each part's gonna represent 10%, and that means I can see the 30% very clearly.

I'll also be able to see 100% very clearly later on.

Now, we know that the 30% is the percentage of the sweets that Alex gave to his brother, and that's equivalent to 15 sweets.

We know therefore that each part has to represent 5 because 15 sweets is being shown with 3 bars, and so 15 divided by 3 is 5, so each part, remember, which is 10%, is worth 5 sweets, and we can fill that in for all the other parts because, remember, all the parts are equal.

Well, this means in order to calculate what the 100% is, I can simply do 10 multiplied by 5, which gives me 50, which means that Alex originally had 50 sweets.

Lucky Alex.

Hmm.

This special offer box contains 600 grammes.

Ah, but because it's on special offer, 20% is free.

Well, hang on.

No, 20% extra is free.

Okay, so that's more than I would normally get.

So how heavy would the box normally be? So it wouldn't normally be 600 grammes.

What would it normally be? Hmm? Well, I can think of that as I have a normal box and because of the special offer, I'm getting more, so I've got an extra 20%, and that's why I've drawn a bit of another box on the top.

And I can see what the normal size would be, and I can see what this extra bit is.

Hmm, remember, I'm told that this whole thing, so my normal plus my extra, is 600 grammes, and that's 120% of what the normal amount would be.

Ah, why do you think I split the normal size bar into five parts? Well done.

It's because the whole bar represents 100%, and so splitting it into five equal parts means each part represents 20%, and that works really well because that's what the extra part had to represent too.

In other words, it's all equal amounts.

So let's calculate how heavy the box is usually.

And now what I've done is I've drawn it as a bar model to make it very simple for myself.

I can now see my 120%.

Remember, that's equivalent to 600 grammes.

I've got here six equal sized bars.

That means in order to find what one bar represents, I need to divide by six, and that means it's 100 grammes, and I can fill that in in each of my bars.

I can see here the 100%.

That's only five of those bars.

So in other words, 5 lots of 100 grammes is 500 grammes, and that's how heavy the box usually is.

Now, Alex and Izzy's teacher is thinking of a number, and they increase it by 20%.

The new number is 162, so what number was Alex and Izzy's teacher thinking of? Well, I can see here that I can represent it with another bar model.

I can see that my new number, my 162, is shown at the top, and the original number, the thing I was thinking of before I increased it, is shown underneath.

Alex says, "To work out the original number, I'm going to do 162 divided by 6, which gives me 27, and then I'm going to multiply that by 5 to get 135." Izzy says, "I'm gonna do 162 divided by 6 is 27, and then I'm going to do 162 take away 27.

Oh, I also get 135." Are both methods appropriate? So in other words, can you justify why they work? Yes, they are.

Alex found 20%, and then he found 100% by multiplying by 5 because if he got 20%, 20% multiplied by 5 will give us 100%.

Izzy found 20%, and then she found the 100% by doing 120% subtract 20%, and this also works.

When might Izzy's method be more useful? Well, in this case, it might have been more useful if we were doing this without a calculator because she doesn't have to multiply, she's instead subtracting, and you may feel that that's easier.

Izzy is saving for a new computer game.

When she saved enough money to buy the game, she noticed the price has increased by 40%.

Oh, poor Izzy.

Oh, look at that.

Her mom's so nice.

She's given her the extra money to buy the game.

How much money had Izzy saved up? Well, what would a bar model look like to represent this situation? Remember, Izzy saved the original amount and then her mum gave her 40% to help her out.

So what would be a sensible partition of our bar model if we want to represent 40%? Well, Alex says, "I'd split the bar model into 10 parts and shade four." And Izzy says, "I'd split it into five parts and shade two." Who do you think is right? That's right, they're both right.

100% divided by 10 and then multiplied by 4 will give us 40%, and 100% divided by 5, well, that gives us 20, and then times it by 2 makes 40, so both work.

So let's actually look at a bar model.

We're gonna go for Izzy's method here.

So we can see here Izzy's method.

Our increased price is up there.

So remember, she decided to break it into five equal pieces where each piece is representing 20%, and then she's got her two extra bars there to represent that increase by 40%.

So there's 140% shown at the top, and we know that's 56 pounds.

So I can see here that I have a total of seven bars, or 56 divided by 7 is 8, so each part there is worth 8 pounds.

I can therefore easily see what the 100% would be.

That's 5 lots of 8, which means the original cost of the game was 40 pounds.

A number is increased by 25%, and the new number is 15.

What was the original number? If you wish, you can sketch a bar model to help you.

Pause the video and have a go now.

Welcome back.

Let's see how you got on.

Well, if you sketched a bar model, I personally would've broken it into four equal parts so that each part's 25%, and that way I can draw a new equal part to be that increase by 25%.

I know that the whole diagram is therefore equal to 15.

I can divide by 5 to work out that each equal part is equal to 3, and I know that four of those equal parts represents the 100%, or my original number, which must have been 12.

Here is a percentage bar which shows the amount of time a set of traffic lights spent on the three different colours.

So remember, traffic lights can be red, amber, or green.

Sometimes of course you could argue that they show amber and green at the same time.

Now, over this period of time, the traffic lights were on amber for a total of four minutes.

Work out how long the lights are on red and green.

Now remember, we're assuming our traffic lights show red, or amber, or green.

They can't show two lights at the same time.

Have you had a go at this? Okay, well let's go through it now and see if the way you've done it is the same as me.

Lucas says, "If we don't know the total amount of time, we can't possibly do this." Oh, so Lucas thinks we couldn't do anything at all there.

Well, I'm really sorry if you spent your time going, "But we can't solve that, Miss," and you were shouting at me on the camera.

Ah, of course, Laura thinks you could have done it.

So if you were telling me you didn't think you could, Laura is saying actually you could have done, and she says that's because we know the percentage and time for amber.

Oh, interesting, Laura.

Let's see if she's right.

Laura says, "We know that the time that was spent on amber was four minutes." What percentage of our total bar is amber? Well, we've got 50% and 45% already shown, so what's remaining must be 5%.

"So we know that four minutes is the same as 5% of our total time." Exactly, Lucas.

5% is four minutes.

"Exactly," says Laura, "and 50% is therefore 10 lots of 5%." So we're gonna multiply 4 by 10, which means that it was on red for 40 minutes.

So how long will the lights be on green for? "Alright," says Lucas, "I get it.

So it's green, it's 9 lots of 4?" And that's exactly right because 45% is 9 lots of 5%.

Well, 9 lots of 4 is 36 minutes, so we spent 36 minutes on the green light.

So here's a new percentage bar which shows the amount of time a different set of traffic lights spent on the different colours.

Work out for how long the lights were showing green.

Pause the video and do this now.

Welcome back.

Well, Lucas says, "I know red is 50%, which means that amber and green must be 60 minutes." Now, he's right because obviously amber plus green was 20% plus 30%, which was 50%, which meant that the red had to be 50% in total, which meant 60 minutes.

So Laura's saying, "Okay, well how can you use that fact therefore?" Can you do that now? Can you help Lucas and Laura work out how long the lights were on green? Now, if you've already done this, just hold on, we'll go through the next bit in a second.

Now remember, red was 50% and that was 60 minutes.

10% of the bar therefore will be our 60 minutes divided by 5, which gives us 12 minutes, or 30% of the bar will be 3 lots of that, so that's 36 minutes, and that's how long we were on green.

We could also solve this type of problem with a double number line.

This can be more useful when the bar cannot be easily partitioned to show the percentage.

Alex has improved his times tables score by 65%.

Well done, Alex.

His score is now 99.

What do you think his score was at the beginning of the year? Now remember, a double number line may be more helpful here.

So I know that his score increased by 65%, so his new score is 165% of his original, which is 99.

I want to find out what the 100% was.

Hmm, so what's that score going to be? My double number line can help me see what steps I have to take to calculate that 100%.

I know that I can multiply any value by the new value I want it to be divided by the original.

In other words, I can turn 165 into 100 by multiplying by 100 over 165.

Remember, our double number lines are showing our multiplicative relationship, so we can use the exact same multiplier with our score.

So 99 multiplied by 100 over 165 tells us that the original score was 60.

Now, Jun increased his score by 15%.

His new score is 92.

What was his original score? Feel free to use double number line to help you.

Pause and do this now.

Welcome back.

Well, the first thing to do is to work out the multiplier I'm going to use.

To calculate my multiplier, I know that I'm going to multiply by a fraction where the denominator is the value I'm starting with, that's 115, and my numerator is the value I want to get to, in this case 100, so I'm multiplying by 100 over 115.

Remember, this is a proportional relationship, so we need to use the same multiplier.

92 multiplied by 100 over 115, it's going to give my original score of 80.

You can also use a ratio table to solve this type of problem.

A special offer bar of chocolate contains an extra 15%.

The special offer bar has a massive 230.

So what's the mass of the normal bar? Well, here's my ratio table.

I know that the mass when it's 230 grammes means that's 115% of the original.

Remember, I want to calculate 100%.

So again, I can find my multiplier, 100 over 115, and then do the same for the mass.

This gives me an original amount of 200 grammes.

Spot the mistake here.

The cost of a TV increases by 22%, and the new price is 512 pounds and 40 pence.

What was the cost of the TV before the price increased? So you can see the working here.

What I want you to do is spot where it's gone wrong.

Pause the video and correct this working now.

Welcome back.

Did you spot the mistake? Well, before we get into that, we could have spotted this answer was wrong even without me telling you.

Let's have a look.

If the price increased to 512 pounds 40, then it must originally have been less than 512 pounds and 40 pence.

And given that I think the original amount is 2,329 pounds and 9 pence, that's got to be wrong because if it's increased, why on Earth did it start off at a bigger amount? It would then have decreased, so that's got to be wrong.

Did you spot where the mistake was? Exactly.

It's there with the 22%.

If I'd increase the cost of the TV by 22%, then the new price is 122% of the original.

That's better.

Of course now my multiplier isn't right, so let's fix our multipliers, and then we can fix the final price.

There we go.

Our original was 420 pounds, and this makes sense now.

It started at 420 pounds and it increased by 22%, so it became more expensive.

That makes more sense.

It's now time for your task.

For question one, four friends live in different places, and they draw bars to show the percentage of days that they experienced different weather on.

Work out how many days were sunny, cloudy, rainy, or stormy for each of the Oak pupils.

Pause and do this now.

Welcome back.

In question two, I'd like you to find the original cost for each of these items, given that the price of the trainers has increased by 10%, so 66 pounds is after that increase, the price of the sunglasses you can see has been increased by 15%, and the price of the car has increased by 17%.

So take those values and work out the original ones, please.

Pause and do this now.

Let's go through some answers.

In question one, I wanted you to work out how many days were sunny, cloudy, rainy, or stormy for each person.

So in a, we needed to work out that we had a total of 20 days here.

So 50% was 10 days, 30% was 6 days, and 20% was 4 days.

In part b, we can see that the sunny days was 20, our cloudy days was 10, our rainy days was 19, and our days that were stormy should have just been one.

And then in c, our sunny amount was for 20 days.

It was cloudy for 12 days, and it was rainy for eight.

And then for d, we were sunny for 120 days, cloudy for 180, rainy for 80, and stormy for a total of 20 days.

In question four, I asked you to find the original cost of the items. Well, the original cost for the trainers was 60 pounds, the original cost for the sunglasses was 32, and the original cost of the car was 8,000 pounds.

It's now time for the second part of today's lesson, and that's on using a multiplier.

I talked about that a little bit in the first part of our lesson, but let's explore it more here.

I think of a number and increase it by 15%.

My new number is 55.

2.

What was my original number? Well, my new number has to include my original number, so that's the 100%, and then my increase, and I increased it by 15%.

Well, that tells me that my new number is 115% of my original.

So what I've done here, I've represented this as it's 115% of my original number, I don't wanna write out the whole phrase original number, so I've just used the letter n to represent that, and that's equivalent to 55.

2.

Now, that of bit is what's so important.

Of can be exchanged from multiplication, and we've seen that a lot.

Finding fractions of an amount, remember that? If I find a half of 20.

We did that by doing a half multiplied by 20.

Same principle here.

So 115% multiplied by my original number is 55.

2.

Now, that 115%, let's explore that.

We can convert that to a decimal very easily, and we can convert number percentage decimals by dividing by 100, so 115% can be written as the decimal 1.

15.

Well, in order to calculate my original value therefore, I can rearrange this equation.

So 55.

2 divided by 1.

15 gives you my original value, which must have been 48.

And this is what we mean by using a multiplier.

I turned my percentage into a decimal, and this allowed me to see the multiplier that I was using and therefore use my multiplier to calculate my original amount.

A special offer bag of sweets has a mass of 336 grammes.

What's the mass of the bag usually? Well, the first thing to do is to say I've got 20% extra here.

This means I have my 100% plus 20% more, so 120% of the original mass, I'm gonna use w to represent that, is equivalent to 336 grammes.

I've turned my percentage here into a decimal so I can see the multiplier.

In other words, to calculate 120%, I took my original amount and I would've multiplied it by 1.

2.

I can therefore rearrange, so do 336 divided by 1.

2, giving me the original mass of the bag, which would've been 280 grammes.

It's now your turn.

So again, a special offer a bag of sweets now has a mass of 210 grammes, so it's still that special offer of 20% extra free.

What is the mass of the bag usually? Pause while you work this out.

Welcome back.

Let's see how you got on.

If 120% of w is 210 grammes, then that tells me that I took my original mass, multiplied by 1.

2, and that would've given me the value of 210 grammes.

I can rearrange this equation to say that the original mass can be found by doing 210 divided by 1.

2, which means the original mass of the bag was 175 grammes.

Well done if you got that right.

It's now time for our final task.

Given the new amount and the percentage increase, I'd like you to find the original amount.

All of the answers are in the grid except for one of them.

For part i, I'd like you to write a percentage question where the original amount is the one value that was left over from the grid.

Pause while you have a go at this.

Welcome back.

In part two, I'd like to complete each statement, and do this in as many different ways as you can.

So increasing a certain amount by a certain percentage gives you 80.

And then in part b, increasing an amount by a certain percentage gives us 250.

Remember, you need to complete these statements in several different ways.

Pause and do this now.

Welcome back.

Question three.

Try to use the multiplier method to answer the following.

So you can see the two questions there.

Try to use the multiplier method that we've looked at in the second part of today's lesson.

Pause and do that now.

Welcome back.

Let's go through some answers.

So you can see here that a, the answer was 325, b, it was 320, c was 340, d was 368, e was 384, f was 350, g was 348, and h was 315.

This means that the value that was left over was 330.

So you had to write a percentage question that gave this answer.

So in other words, if my original value was 330 and I increased it by 35%, then my new amount would be 445.

5.

Remember, for i, you could have used a variety of values there.

In question two, we said remember, complete the statement in several different ways.

Here are some examples.

Remember, you might have done something different, and it's absolutely fine if you have.

In question three, we said to try to use the multiplier method to answer the following.

So a social media manager earns 34,980 pounds a year, and this includes a pay rise of 6%.

What were they earning before? Well, that means that my original wage was multiplied by 1.

06 in order to get to 34,980, so if I divide my new amount by 1.

06, then I'll get my original wage, which was 33,000 pounds.

And in part b, you wanted to work out the original cost of a ticket last year, so that was before the price increase of 28%.

I need to take the new cost of the ticket, so that's 89 pounds 60, and divide by the multiplier, which will be 1.

28.

This means the original ticket would've cost 70 pounds.

Let's sum up what we've learned today.

Bar models, double number lines, ratio tables, and multipliers can be used to find original amounts.

Of can often be exchanged for multiplication in our maths problems. And remember, there are certain words that indicate there's been a percentage increase, and those sorts of words are profit, extra, and interest.

Well done.

You've worked really well today.

I look forward to seeing you in one of our future lessons.