Problem 1 - Palindromic sales
Problem 1 is all about palindromes: words, phrases or numbers that read the same forwards or backwards.
Maths teacher Chris Smith is here to explain. WOW!
The Maths Week Scotland Daily Challenges have been set by the Scottish Mathematical Council.
Chris Smith - Palindromes are numbers, or words, that are the same front to back, or should that be back to front?
This problem is about percentages and the numbers are palindromic. But it’s not just the numbers in the question that are palindromes.
Mr Bob Hannah is the owner of a gadget shop called Universal Palindromes Unlimited - UPU for short.
Bob has a sale on. At first he offered a WOW discount of 55% off. (55 is a palindrome).
Now he’s reduced the prices further by a Manager Special amount of £10.01. (You guessed it - another palindrome).
Otto bought a nice set of headphones from Bob for a very reasonable (and palindromic) £44.44.
But how much cash would Otto have needed to part with if he’d bought the headphones at full price with no discount?
You know Bob Hannah loves palindromes. Do you think the answer might be a palindrome too?
Remember, you know the final price and you know the discounts, so you’re going to need to work backwards to see if you can find the original price.
Can you work it out?
So here's the challenge:
Bob Hannah offered a WOW discount of 55% on all items.Then he reduced the prices by a further £10.01.
Otto bought a pair of headphones for £44.44.What would he have paid for them if neither of the discounts had been offered?
Need a hint?
Bob Hannah loves palindromes. Do you think the answer might be a palindrome too?
You know the final price and you know the discounts, so you’re going to need to work backwards to find the original price.
Solution
Worked out the answer? Here's how you can do it.
Chris Smith: How did you get on?
Did you spot some more palindromes?
The key to this problem is adding the discounts back onto the price in reverse order.
Kenny bought the headphones at a discounted palindromic price of £44.44.
First add Bob’s Special Price discount of £10.01.
£44.44 plus £10.01 equals £54.45.
Did you notice £54.45 is also a palindrome
We know the WOW discount price took 55% off the full price.
So £54.45 is what is left when you take 55% away from 100%. It’s 45% of the full price.
We know that when we say 45 per cent of a number what we really mean is multiply it by 45 divided by 100. But what was that original number?
To reverse the percentage, we need to divide instead of multiply. Dividing by a fraction is the same as multiplying by its reciprocal.
So we can just flip the fraction upside down.
We multiply £54.45 by 100/45
Giving a final full price of £121
You guessed it – another palindrome!
Palindromes are the same forwards and backwards but you definitely need to work backwards to solve this problem.
You know the final price is £44.44.
First you need to work out the price before the second discount.
The second discount took £10.01 off the price. So you need to add this back on to £44.44.
£44.44 + £10.01 = £54.45
To work out the original price you now need to add the WOW discount back on.
The WOW discount took 55% off. How much does that leave?
100% - 55% = 45%
So that means the reduced price of £54.45 was 45% of the full price.
You need to work out 100% and you can do this by first working out 1%.
1% = 45% ÷ 45
= £54.45 ÷ 45
= £1.21
The full price is 100 times this.
100 X £1.21 = £121
The original price of Otto's headphones was £121.
More on Problem solving
Find out more by working through a topic
- count16 of 21
- count17 of 21
- count18 of 21
- count19 of 21
