Problem 4 - Shopping spree
Problem 4 is all about working out how much money has been spent on a family shopping trip.
Maths teacher Chris Smith and pupils from Grange Academy are here to explain.
The Maths Week Scotland Daily Challenges have been set by the Scottish Mathematical Council.
CHRIS: In this problem, we're going on a shopping spree and we need to work out how much Tam McTavish spent.
All eleven members of the McTavish family have gone on a shopping trip to Killiemuir.
Ten of them spent exactly twenty pounds each. But Tam was extravagant. He spent ten pounds more than the average for the whole family.
How much did Tam spend?
Explain your answer.
Now, ten of them spend exactly twenty pounds when they're out shopping but Tam, he's a bit extravagant and spends more.
PUPIL: Remember Tam spent ten pounds more than average. Not ten pounds more than everyone else.
CHRIS: Tam spends ten pound more than the average, right.
PUPIL: It might help how to work out a formula to find the total money the family spent.
PUPIL: You could try trial and improvement to get to the right answer.
CHRIS: You think that's going to work?
Because this is what the average is going to be. Is that even going to be a whole number?
PUPIL: Try your best and see if you can crack it.
So here's the challenge:
All eleven members of the McTavish family have gone on a shopping trip to Killiemuir.
Ten of them spent exactly £20 each but Tam was extravagant. He spent £10 more than the average for the whole family.
How much did Tam spend?
Need a hint?
- Remember Tam spent £10 more than the average – not £10 more than everyone else.
- It might help to work out a formula to find the total money the family spent.
- You could try trial and improvement to get to the right answer.
Solution
Worked out the answer? Here's how you can do it.
Did you work out much Tam McTavish spent on his family shopping spree? You could've used algebra to work this one out.
Let's take a look.
We know that there were eleven people on this shopping spree and ten of them spent twenty pounds.
We also know that Tam spent ten pounds more than the average for the whole family.
Suppose that Tam spent X pounds then the total spend is ten times twenty plus X.
That equals two hundred plus X.
So the average amount spent is one eleventh of two hundred plus X, so that means X equals ten plus one eleventh multiplied by two hundred plus X.
We can multiply both sides of the equation by eleven. So that eleven X equals one hundred and ten plus two hundred plus X.
We can then take X away from both sides of the equation, leaving us with ten X equals three hundred and ten.
If we divide both sides by ten, we can find out what the value of X is and that is thirty one.
Tam spent thirty one pounds on the family shopping spree.
Well done if you figured it out.
Here is what we know:
- \(11\) people went on the shopping trip.
- \(10\) people spend \(£20\) each.
- Tam spent \(£10\) more than the average for the whole family.
We can use algebra to work out the answer
Step one
If we say Tam spent \(x\), we can work out an expression that shows the total that was spent.
\(\text{total} = 10 \times 20 + x\)
\( = 200 + x\)
Step two
We know that Tam spent \(£10\) more than the average for the whole family. We can write this as:
\(x = 10 + \text{average}\)
Step three
If the total spend was \(£200 + x\), then the average is \(\frac{1}{11}\) of this, as there were \(11\) people on the trip.
\(\text{average} = \frac{1}{11} (200 + x)\)
We can put this into our equation.
\(x = 10 + \frac{1}{11} (200 + x)\)
Step four
Simplify the equation by multiplying both sides by \(11\).
\(11x = 110 + 200 + x\)
Step five
Take \(x\) from each side.
\(10x = 110 +200\)
\( = 310\)
Step six
Divide both sides by \(10\)
\(x = 31\)
So we have worked out that Tam spent £31.
Maths Week Scotland 2023. list
Try out all the daily challenges from Maths Week Scotland 2023.
Maths Week Scotland 2022. list
Try out all the daily challenges from Maths Week Scotland 2022.
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