Multiplication mistake
Jane accidentally multiplied by 54 instead of 45, and her answer was 198 too big. What number did she multiply 54 by?
Age
11 to 14
Challenge level
Problem
Jane made a mistake when writing down a multiplication, and she multiplied by 54 instead of 45.
Her answer was 198 more than it should have been.
What number did she multiply 54 by?
This problem is adapted from the World Mathematics Championships
Student Solutions
Answer: 22
Using algebra
Let the number that Jane multiplied by be called $n$. Then $54n$ is $198$ more than $45n$.
So $54n=45n+198$, so $54n-45n=198$, so $9n=198$, so $n=198\div9=22$.
Trying different numbers and using proportion
10: 54$\times$10 = 540, 45$\times$10 = 450 difference = 90 which is 9$\times$10
20: 54$\times$20 = 1080, 45$\times$20 = 900 difference = 180 which is 9$\times$20
Need difference of 198 = 9$\times$22, so try 22
22: 54$\times$22 = 1188, 45$\times$20 = 990 difference = 192
Why is it 9 times the number? Because 54 $-$ 45 = 9