### Inequalities

A bag contains 12 marbles. There are more red than green but green and blue together exceed the reds. The total of yellow and green marbles is more than the total of red and blue. How many of each colour there are in the bag?

### What's it Worth?

There are lots of different methods to find out what the shapes are worth - how many can you find?

### Turnips

Baldrick could buy 6 parsnips and 7 turnips, or 8 parsnips and 4 turnips. How many parsnips could he buy?

# Multiplication Mistake

##### Age 11 to 14 Short Challenge Level:

Working it out by trying different numbers
54$\times$10 = 540 and 45$\times$10 = 450. The difference between 540 and 450 is 90, which is much smaller than 198. So the number must have been much greater than 10.

54$\times$20 = 1080 and 45$\times$20 = 900. The difference between 1080 and 900 is 180, which is a bit smaller than 198. So the number must have been a bit more than 20.

54$\times$21 = 1134 and 45$\times$21 = 945. The difference between 1134 and 945 is 189, which is a bit smaller than 198. So the number must have been a bit more than 21.

54$\times$22 = 1188 and 45$\times$21 = 990. The difference between 1188 and 990 is 198! So the number must have been 22.

Trying different numbers and using proportion
54$\times$10 = 540 and 45$\times$10 = 450. The difference between 540 and 450 is 90, which is much smaller than 198. So the number must have been much greater than 10.

54$\times$20 = 1080 and 45$\times$20 = 900. The difference between 1080 and 900 is 180, which is a bit smaller than 198. So the number must have been a bit more than 20.

Using 20 instead of 10, the difference doubled, so the difference increased by the same proportion as the numbers we used. 20 gave us a difference of 180, which is 18 less than the difference we wanted. 18 is 10% of 180, so to increase the difference by 10%, we should multiply by a number that is 10% more than 20.

10% of 20 is 2, so Jane must have multiplied by 22.

To check, 54$\times$22 = 1188, 45$\times$21 = 990, and the difference between 1188 and 990 is 198!

Using the laws of multiplication
54 is 45 + 9. So multiplying by 54 is the same as multiplying by 45 and multiplying by 9, and adding the answers together.

So when Jane multiplied by 54 instead of 45, she got 9 extra lots of the number she multiplied by.

So 9$\times\text{the number}$ must be 198, so $\text{the number}$ must have been 198$\div$9 = 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 $n=198\div9=22$.

You can find more short problems, arranged by curriculum topic, in our short problems collection.