Lesson video

Hi, I'm Mrs. Warehouse and welcome to today's lesson on finding the original amount after a decrease.

Now this lesson falls in the unit, Understanding Multiplicative Relationships, Percentages and Proportionality.

I hope you enjoyed today's lesson.

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 decrease.

Hmm, wonder what I mean by that? Well, let's have a look at this lesson and find out.

Here is a key word that we're gonna be using today, and it's the word reciprocal.

Now, you might be very familiar with this word, and if you are, that's great, but if you are not, feel free to pause the video.

Have a read of the definition now so that you are ready and you can use the word comfortably during the lesson.

Our lesson today is broken into three parts and we're going to start with Part One, which is finding the original value.

Oh, look at that cat doesn't it look cute? Ah, it's Alex's cat and it goes to the vet once a year probably for a checkup or booster jabs.

The vet tells Alex that his cat has lost 30% of its mass.

Oh, okay.

So the cat's lost here, 30% of its mass.

Ah, it means it's got lighter.

Okay, the cat now has a mass of 4.

2 kilos.

What was the mass of Alex's cat last year, so before it lost some of this mass? I wonder how we're gonna work that out.

Well, let's start with our trusty bar model.

We've used these a lot and they've been really helpful for understanding how we do percentage increase, decrease.

We've used them for ratios.

In fact, they've been helpful in a lot of bits of maths.

So let's see how we're gonna use it here.

We're going to show that the mass loss falls at the end here.

In other words, it's the final three parts you can see that I've coloured them in.

And then the new mass, i.

e.

, what the cat weighs now must be the remaining part.

So the original mass will be the mass the cat is at the moment, plus the mass that it lost.

Well, the cat now has a mass of 4.

2 kilos, so I can put that on my diagram.

I know that if I divide it by seven, that 0.

6 kilos is what I'll get.

And that must be what each equal part of the bar below represents.

So you can see I filled in the 0.

6 kilos.

Well, because they're equal parts, the mass lost, those three parts also have to be 0.

6.

Well, I can now say that the cat originally had a mass of six kilos because I have 10 lots of 0.

60.

<v ->Oh hi Lucas.

</v> <v ->Hi Sophia.

</v> Let's see what they're talking about.

Oh, Lucas is saying he loves Sophia's new hoodie and he wonders if he's got enough money to get one too.

So when Sophia originally bought the hoodie, it was 41 pounds, 25 pence, but that was when the shop had a 25% sale on, smart girl, but that's finished now.

Hmm, Lucas is like, well what's it gonna cost now? Luckily we can work that out.

So we've got our bar model here.

Can you see we've broken it into four equal parts.

Why do you think that is? That's right, we're talking about a 25% sale.

Four lots of 25 makes a hundred.

So it makes sense if my original value was a hundred percent, then breaking it into parts that are 25% will be useful to me.

So at the end, I've got the amount that Sophia saved.

Remember she bought it for less than it originally cost.

So there's our amount saved.

The rest therefore must be the sale price and that's what Sophia actually paid for the hoodie.

So that'll be 41 pounds, 25 pence.

And the original price, remember, is what Lucas wants to find out 'cause he wants to know what the hoodie would cost him if he bought it today.

So we've got our sale price written up.

We're gonna divide it by three because we have three equal parts, which means each part is worth 13 pounds and 75 pence, and you can see I filled that in.

But we know that's also the amount that Sophia saved because the amount saved is an equal part to the three parts I have for the amount that it actually cost her.

That means to work out the original price, I can simply do 13 pounds 75 times four.

And that means the hoodie now costs 55 pounds.

Right time for a quick check.

You pay 36 pounds for these trainers because they're in a sale.

I'd like you to use the bar model to work out the original price of the trainers, so if you bought them and they weren't in a sale.

Pause the video and do that now.

Welcome back.

Let's see how you got on.

Well, first things first.

We know that the bar underneath, so the entire bar is the original price.

We can put up at the top our sale price, and then we're going to divide by four because there are four equal parts here.

That means each part is worth nine and therefore the amount saved was nine pounds.

So to work out the original price, I simply need to do nine times five, which is 45 pounds.

We can 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.

So the cost of a car depreciates by 16% in the first year.

Yeah, I don't fancy trying to split my bar up into equal parts where one part represents 16% because 16 is not a factor of a hundred.

So double number line it is, we now know the car is worth 12,600 pounds.

So how much did the car cost when it was first purchased? What percentage of the original cost is the car now worth? We can work that out quite easily.

If the car's original cost was a hundred percent and it's depreciated by 16%, then the car is now 84% of what it was originally worth.

So given it's now worth 84% of the original cost, we can use our double number line.

Notice that I've put the 84 on the percentage line a little further down, and that's obviously because a hundred percent is going at the top.

So what cost goes with the 84%? That's right, it's the 12,600.

Now remember, we want to work out what a hundred percent would be.

So what's gonna go there? And this is where we can work out a multiplier.

How do you move from 84 to a hundred? What do you need to multiply by? That's right.

You are multiplying by a hundred and dividing by 84.

In other words, you're multiplying by a hundred over 84.

Remember, we must do the same thing on the cost line because what we have here is a multiplicative relationship and these lines are scaled.

So multiplying 12,600 by a hundred over 84 gives us our original cost, which was 15,000 pounds.

A reduced sugar chocolate bar contains 30% less sugar than a regular bar.

The reduced sugar bar contains 36.

8 grammes of sugar.

So how much sugar is in a regular bar of chocolate? What we could use a ratio table.

Remember this is one of our other representations that can be really helpful when we are working with percentages.

So we've put in the 36.

8 grammes because that's the information that came from the question.

But what percentage of the original bar does the 36.

8 represent? Hmm, well let's read the question again.

It contains at 30% less sugar.

Well, if the original bar was a hundred percent and this is 30% less, then this percentage must be 70.

Now I can think about how I'm going to get from my 70% back to the original of a hundred percent.

What would I multiply by? That's right, I multiply by a hundred over 70.

Remember, I need to do the same to both sides of my table because here I have a multiplicative relationship.

So we multiplied by a hundred over 70, which means there was 52.

6 grammes of sugar in a regular bar of chocolate.

Now I've rounded that to one decimal place.

I think that's quite sensible rounding.

Your turn.

Using any of the three different methods, so whether you want to use a bar model, double number line or ratio table, I'd like you to answer this question.

A low packet of crisps contains 0.

17 grammes of salt.

This packet contains 45% less salt than a regular bag does.

So how much salt is in a regular bag? Pause the video and have a go at this question now.

Welcome back.

Let's see what method you went for.

I'm gonna share this in a ratio table, but remember you could have picked a different method and that's absolutely fine.

I know that the pack contains 45% less salt.

So if the original was a hundred and this is 45% lower, that means it has to represent 55% of the original.

So that's why I filled in my first row.

In order to move to what a hundred percent is, I need to multiply by a hundred over 55, same to both sides, of course.

That means that a regular bag contains 0.

31 grammes of salt.

And again, I've rounded that to two decimal places in this case because the value in the question was to two decimal places.

So I've kept to the same level of accuracy.

It's now time for your task.

In Part 1A, I've said a tee shirt costs 21 pounds in a sale offering a 25% discount.

What was the cost of the tee shirt before the discount? Now at the top I have said you can use any preferred method to answer the following questions and you need to give your answers to an appropriate degree of accuracy.

So think about the context and what level of rounding you're going to want to do.

Pause the video while you complete 1A.

And 1B.

A car's value depreciates by 22% in the first year.

It's now worth 11,544 pounds.

What was the cost of the car originally? Pause the video while you have a go.

And then 1C.

A pack of crisp contains 33% less fat.

This lowfat packet contains 5.

3 grammes of fat.

How much fat is in a regular packet? Pause and have a go.

Time to go through your answers now.

So for 1A, I asked for the original cost of the tee shirt before the discount and you should have got the price of the tee shirt before the discount is 28 pounds.

A car's valued depreciated by 22%.

What was the cost of the car originally? And you should have got 14,800 pounds.

And then for part C, how much fat is in a regular packet? You should have reached 7.

9 grammes of fat.

Time for the second part of our lesson, and this is going to be using a multiplier.

Aisha buys her cinema ticket online.

This saves her 6% of the usual cost of the ticket.

My ticket costs me seven pounds and five pence.

Oh, Jacob would like to go but he can't get his ticket online.

That means no discount.

What's his ticket gonna cost? Well, what percentage of the usual ticket price did Aisha should pay? Well, she had a 6% saving, which meant she paid 94% of the original cost.

So 94% of our original cost is seven pounds and five pence.

Notice that instead of writing original cost or usual cost of the ticket, I've just written C.

And that's because I like notation.

It means I don't have to write all these words out, it's just easier.

So we're gonna use that I'm gonna work out the original cost.

So in other words, what does c stand for? Now remember of, can be exchanged for multiplication.

In other words, I'm multiplying here, but I need to turn that percentage into a decimal so that I've got my multiplier.

So 94%, we're gonna convert that to a multiplier and we're gonna do that by turning it to a decimal.

Now we can do that easily.

We turn a percentage into a decimal simply by dividing by a hundred.

So 94% divided by a hundred is 0.

94.

So what we've actually got is we've taken our statement and we've been able to write it as an equation.

0.

94 multiplied by the original cost will give us the cost that Aisha paid, which was seven pounds and five pence.

We can rearrange that to find the usual cost of a ticket.

So seven pounds and five pence divided by 0.

94 means the usual cost of a ticket is 7 pounds 50.

Let's consider this one.

The temperature between midday and midnight has fallen by 67%.

The temperature at midnight was three degrees centigrade.

What was the temperature at midday to the nearest degree? Well, what percentage of the midday temperature is the midnight temperature? Remember it fell by 67%.

Well that means it's 33% of the original.

So 33% of our original temperature is equal to three.

And again, we can now rewrite this using a multiplier.

So remember the of gets exchanged so it becomes multiplication and the 33% we're going to divide by a hundred to get the decimal 0.

33.

We simply rearrange and then do our division.

So the original temperature was nine degrees centigrade.

Time for a quick check.

An online store sells a TV for 393 pounds and 60 pence.

This is 18% lower than a shop on the high street.

How much does the TV cost on the high street? Pause the video and have a go now.

Welcome back.

Did you spot right gone wrong? Yeah, exactly.

If you've looked straight away, you've gone, "Hang on a second, you said this was 18% lower.

That doesn't mean it's 18% of the original cost.

That needs to be fixed straight away.

So that 18% lower, that was the key bit I should have got here I should have got here or I should have worked out, a hundred percent take away the 18% so that I know the percentage of the original cost is in fact 82%.

Now I can start working this out.

So that's 82%, hang on, divided by c.

Hmm, I'm not sure that of normally represents division.

That's right, did you spot that should have been multiplication.

Gosh, we made loads of mistakes here.

Okay, what about that multiplier? Okay, I've changed that through, so that multiplier should be 0.

82 'cause it's 82% of the original cost, or rather the cost on the high street.

And then I've gotta change the other side so when I rearranged it was correct and that multiplication needs to be a division, now I can change my final value.

And finally this question is right, the TV costs 480 pounds on the high street.

Now it's time for your second task.

Given the new amount and the percentage decrease, find the original amount.

Now all of the answers are in the grid except one.

For Part i, you need to write a percentage question that gives the leftover amount as your final answer.

In other words, whatever amount gets left, this would've been the original amount.

You need to come up with a percentage decrease and a new amount that works with that value.

Pause the video and have a go now.

Welcome back.

Let's go through our answers.

So for a, should have 350 b, 340 c, 320 d, 348 e, 384 f, 368 g, 330, and h, 315.

Now that meant the value you were left over with was 325.

So for example, if I have 325 and I reduce or decrease that value by 35%, my new amount would be 211.

25.

Now remember, for i, that's just an example, there are loads you could have come up with that would work.

Time for the final part of today's lesson and that's on mixed reverse percentage problems. Reverse percentage problems are problems where you know the new value and you need to find the original value.

So for example, a special offer box of cereal contains 375 grammes of cereal.

This is 25% more than the usual box.

How heavy is the usual box? Or we could say Izzy buys a bag from a shop advertising a 10% sale.

She pays 22 pounds 50 for the bag.

What was the original cost? So can you see here, we know there's either been an increase or a reduction and we need to work back to find the original value.

Now in the questions like this, you have to determine whether it's an increase or a decrease.

Sometimes the question makes it really obvious and will tell you.

And other times it will use key words instead.

And we have to understand what those words mean.

Let's consider this one.

There are 72 apples in this special offer box.

How many apples are in a normal box? Now Izzy says, I think this is one where there's been a reduction.

So I think I should do a hundred percent, take away the 20% and that means I want 80% of this amount.

And that's what she's done.

She's done 72 divided by 0.

8 has to be 90.

So she thinks the original number of apples was 90 and the special offer means this got reduced to 72.

Does that seem right to you? Yeah, it doesn't does it? Izzy subtracted 20% from a hundred percent but she should have added it on because the box has an extra 20% and we could tell she'd gone wrong because she got 90 apples and the special offer was for 72, but she's supposed to have got more for free.

That definitely was the wrong way around.

And that's a great way to check if you think you've done this right or not.

So that's how we know Izzy's answer was incorrect.

Let's try that again then.

So we know the number of apples in the special offer box.

So the usual box plus the extra free is 72.

So that's our original amount, a hundred percent add the 20% that was extra, means that's 72.

So 120% of the original number of apples is 72.

So remember I can write that statement mathematically as 1.

2 multiplied by a is equal to 72.

I can rearrange and that tells me that the original amount, so in other words there are 60 apples in a normal box.

So for each of the following, we're gonna decide if we need to subtract the percentage from a hundred or add the percentage on.

The height of sunflower increases by 10% over a week and now measures 1.

2.

How tall was the sunflower last week? Can you see that in the question? It says increases.

This means we are adding on.

What about this one? The water level and pond drops by 5%.

The level is currently 45 centimetres.

What was the level before the drop? So do you think we've had an increase here or a decrease? Did you spot the word drops? Exactly, this is a decrease.

What we've now got is 95% of the original because the water level dropped.

A puppy gains 8% of its birth maths over the first three months.

It now weighs six kilos.

How much did it weigh at birth? So is this an increase or a decrease question? Ah, you spotted the gains, that's right.

We actually have an increase, so it'll be a hundred percent add the 8%.

The cost of a camera increases by 15% to 276 pounds.

What was the cost of the camera originally? I'm gonna show you three ratio tables and you have to decide which one is the correct one for this problem.

Pause the video while you have a think.

Welcome back.

Which one did you pick? Remember we've got an increase here.

So we know that the 276 pounds is the original amount, a hundred percent, increased by 15%.

So we've added on 15%, which means it's the middle table.

The first table says that 276 pounds is 15% and the final table represents a decrease of 15%, not an increase.

So what was the cost of the camera originally? Pause and work this out.

Welcome back.

What did you multiply by? You should have said it's a hundred over 115.

And then the same for the other side, which means the cost of the camera was originally 240 pounds.

So let's think about some of those key words.

A shop offers a discount for bulk buyers of 5% discount.

Discount, that tells us we have a reduction.

A bank gives 2.

3% interest on a savings account.

What's the key word that's helping us there? That's right, interest.

We've had an increase.

A house is sold at a loss of 12%, there's the word loss.

Pause and work out what the other three words are for the remaining three statements.

Say statement four should have found the word depreciates, meaning it's reduced.

In statement five, we should have had extra.

So we've had an increase.

And then lastly, profit, so we've had an increase.

We're now gonna sort these into whether they're increases or decreases.

So remember I've just gone through them with you, so this should be fairly straightforward.

Pause the video and sort them now.

Welcome back, let's check you've got 'em in the right places.

So let's sort these all out.

There we go, Excellent.

So decrease are words like discount, loss and depreciates.

Increase would be interest, extra and profit.

So when you spot those words, they're telling you what's happened in the question.

Let's do a quick check.

Sam's aunt buys a vase and sells it for a profit of 45%.

She sells it for 232 pounds.

How much did she originally pay for it? Pause, spot and correct any mistakes you can find.

Welcome back.

What did you find that had gone wrong? Right, this is a profit.

In other words, the amount we sold it for represents an increase from the original amount.

So that 55% is not right.

It should be 145 and that means our multipliers are wrong.

It should be a hundred over 145 and that means our final value should be 160 pounds.

So she originally paid 160 pounds and she sold it for 232.

Wow, 72 pounds profit, nice one Sam's aunt! Time for your final task.

For question one, decide whether each of the following questions indicates a decrease or an increase.

Pause and do this now.

Question two.

So now you've made your decision about whether it's an increase or a decrease question.

I'd like you to use your preferred method to find the original amounts for each question.

Pause and do that now.

Welcome back.

Let's go through some answers.

So in question 1, what I've done is I've shown which words were telling me if it was a decrease or an increase question.

So loss, decrease, extra, increase, depreciates, decrease, interest, increase, and discounted shows us it's a decrease.

Let's look at question 2 now, where you had to find the original amounts.

So for 2a, the original amount was 72 pounds.

In b, we had 16 apples originally.

C, the original value of the motorbike was 8,525 pounds.

And in d, the savings account started with 600 pounds in it.

And in E, the laptop's original cost would've been 260 pounds.

So let's sum up what we've learned today.

Bar models, double number lines, ratio tables and multipliers can be used to find the original amount or can be exchanged fr multiplication.

A multiplier is the decimal equivalent of a percentage.

And keywords such as discount, loss, depreciation usually indicate a decrease.

Whilst keywords such as profit, extra and interest usually indicate an increase.

It's so important that we fully understand the context when we're trying to work out, if we are dealing with an increase or a decreased type of question that can help us work out the correct approach.

Well done, you've worked really hard today and I hope you enjoyed the lesson.

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