I have the following question in a past paper. I think I answered it correctly but am not sure and would appreciate someone to let me know if it is indeed correct:
Mrs. Alishaw owns many books on a number of subjects. The ratio of Maths books to History books is 4:11. The ratio of Maths and Physics books combined to History books is 7:13. What is the minimum number of books that Mrs Alishaw owns?
This is what I did:
Since the number of history books will remain the same, it must have been simplified to 11 in one case and 13 in the other (unless she has a fraction of a book, which is unlikely) as they are both prime numbers. Thus, I found the lowest common multiple (143) and added that to the number of Maths and Physics Books combined (multiplied by 11, 143/13, to fit the ratio), 77, to get 220.
Thank you in advance for your help
Sent from a new poster (n00b)
You are correct.
The key is indeed assuming integer values for $m$, $h$, and $p$, and working from the divisibility relations that come out of the ratios you are given:
$4 \mid (11m) \implies 4 \mid m$
$7 \mid (13(m+p)) \implies 7 \mid (m+p)$
$11 \mid (4h) \implies 11 \mid h$
$13 \mid (7h) \implies 13 \mid h$.