# Multiplication Mistake

Jane accidentally multiplied by 54 instead of 45, and her answer was 198 too big. What number did she multiply 54 by?

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*

**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*