Challenge 1 - Currency
Challenge 1 is all about currency and different exchange rates.
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.
Mr Smith: This problem is all about money.
I have one US dollar, one Australian dollar and one New Zealand dollar.
The US dollar and the Australian dollar are together worth £1.27.
The Australian dollar and the New Zealand dollar are together worth 95p.
The US dollar and the New Zealand dollar are together worth £1.18.
What would 100 US dollars be worth?
Explain your answer.
Pupil: Before you find the value of 100 dollars, you might want to find the value of one.
Pupil: Using algebra might make this easier to work through.
Pupil: You'll need to use all three statements to find your answer.
Pupil: Give it a spin.
So here's the challenge:
This problem is all about currency.
We have one US dollar, one Australian dollar and one New Zealand dollar.
The US dollar and the Australian dollar are together worth £1.27.
The Australian dollar and the New Zealand dollar are together worth 95p.
The US dollar and the New Zealand dollar are together worth £1.18.
How much would 100 US dollars be worth?
Need a hint?
Before you find the value of 100 US dollars, you might want to find the value of one dollar.
Using algebra might make this easier to work through.
You'll need to use all three statements to find your answer.
Solution
Worked out the answer? Here's how you can do it.
Did you work out how much 100 US dollars would be worth?
Here’s one way to work out the answer.
If we change the values of the three different dollars into letters, we can create some equations. So US dollars can be 𝑥, Australian dollars can be 𝑦, and New Zealand dollars can be 𝑧.
This gives us these three equations:
𝑥 + 𝑦 = 127
𝑦 + 𝑧 = 95
𝑥 + 𝑧 = 118.
We need to use the equations together to find out the different values.
If we take the second equation away from the first, we get 𝑥 + 𝑦 = 27, subtract 𝑦 + 𝑧 = 95, which gives us 𝑥 − 𝑧 = 32.
If we now subtract this from our third equation 𝑥 + 𝑧 = 118, we get 2𝑧 = 86. Therefore 𝑧 = 43.
And now we know the value of 𝑧, we can substitute that into 𝑥 – 𝑧 = 32 , and that gives us 𝑥 = 75.
So one US Dollar is worth 75 pence.
And 100 US Dollars are worth 100 × 75p = £75.
Alternatively, you could add all three equations together to get 2𝑥 + 2𝑦 + 2𝑧 = 340; which means 𝑥 + 𝑦 + 𝑧 = 170.
The second equation shows us that 𝑦 + 𝑧 = 95, so 𝑥 + 95 = 170. That means 𝑥 = 75.
And again 100𝑥, or 100 US dollars = £75.
Great job if you aced this challenge.
Step 1
If we change the values of the three different dollars into letters, we can create some equations.
So US dollars can be x, Australian dollars can be y, and New Zealand dollars can be z.
Step 2
This give us these three equations:
\(x + y = 127\)
\(y + z = \ \ 95\)
\(x + z = 118\)
Step 3
We need to use the equations together to find out the different values.
If we take the second equation away from the first, we get:
\(\quad x + y\quad \quad = 127\)
\(\underline{-\quad \quad y + z \ = \ \ 95}\)
\(\quad x \quad \quad − z = \ \ 32\)
Step 4
We can now take this new equation away from the third equation:
\(\quad \ \ x \quad \ \ \ + z = 118\)
\(\underline{\ - x \quad \ \ \ - z = \ \ 32}\)
\(\underline{\quad \quad \quad \quad \ \ 2z = \ \ 86}\)
Divide both sides of this equation to find \(z\):
\(z = 43\)
Step 5
We can now substitute the value of \(z\) into \(x – z = 32\):
\(\quad \ x \ – \ \ z = 32\)
\(\Rightarrow x \ – \ 43 = 32\)
\(\Rightarrow x = 75\)
So \(1\ US$ = 75p\)
Step 6
And to find the value of one hundred US dollars we multiply by 100:
\(100 × 75p = £75\)
One hundred US dollars are worth £75.
Alternative solution
Another way to solve this is to add all three equations together:
Step 1
\(\quad \ x + y\quad \quad = 127\)
\(\quad \quad \quad \ y + z \ = \ \ 95\)
\(\underline{+ \ x \quad \quad \ + z = 118}\)
\(\ \ 2x + 2y + 2z = 340\)
Step 2
We can divide both sides of this equation by two:
\(x + y + z = 170\)
Step 3
Substitute in the second equation (\(y + z = 95\)) to find the value of \(x\):
\(\quad \ x + y + z = 170\)
\(\quad \ x + 95 \quad = 170\)
\(\Rightarrow x = 75\)
So \(1\ US$ = 75p\).
Step 4
Now multiply by \(100\) to find the value of \(100 \ US$\):
\(100 × 75p = £75\)
One hundred US dollars are worth £75.
Maths problem solving. collection
Try out Maths Week Scotland challenges from previous years.
Maths Week Scotland. External Link
Find more maths challenges and activities at the Maths Week Scotland website.
