There are 3 crates of apples A, B and C. The ratio of the number of apples in crate A to crate B to crate C is 8 : 5 : 3. David removed 42 apples from crate A and placed them into crate C. as such, there are 2 more apples in crate C than crate B. How many apples are there altogether?

2. HINT: 3k + 42 = 5k + 2

4. Originally we must have integers in the ratio $\displaystyle 8k:5k:3k$ clearly we can't have fractional apples.

you are adding 42 to the crate that is in the ratio 3.
And then you are told this is 2 more than the amount in the crate of ratio 5.

So that means for some integer k, we need the equation posted above to be satisfied.

$\displaystyle 42+3k=5k+2\Rightarrow 40=2k \Rightarrow k=20$

That means 20 is this common ratio we are looking for.

Crate A $\displaystyle =20\cdot 8=160$
Crate B $\displaystyle =20\cdot 5=100$
Crate C $\displaystyle =20\cdot 3=60$
Add these up to get 220 apples.

You check and see that if you add 42 to 60 you get 102 which is 2 more than 100 and these crates are in the proper ratios, thus the answer is correct. Well done wilmer, hope you don't mind me jumping in here, I just saw that you were not signed in and didnt want gwen to wait for a response.

5. Thank You very much, Gamma and Wilmer.

6. Originally Posted by gwen
There are 3 crates of apples A, B and C. The ratio of the number of apples in crate A to crate B to crate C is 8 : 5 : 3. David removed 42 apples from crate A and placed them into crate C. as such, there are 2 more apples in crate C than crate B. How many apples are there altogether?
You are given the ratio, and told that there are at least 42 items in A (else how could 42 be remove, right?), so one way to start might be to list triples in the given ratio, with the first value being 42 or larger. We can safely assume that we are dealing with whole numbers, so:

48 : 30 : 18
56 : 35 : 21
64 : 40 : 24
72 : 45 : 27
80 : 50 : 30
88 : 55 : 33

...and so forth. We can also add 42 to the third number, and compare with the second number, because we know we want the third to be two greater than the second:

48 : 30 : 18 => 30 : 60 (60 - 30 = 30: too large)
56 : 35 : 21 => 35 : 63 (63 - 35 = 28: too large)
64 : 40 : 24 => 40 : 66 (66 - 40 = 26: too large)
72 : 45 : 27 => 45 : 69 (69 - 45 = 24: too large)
80 : 50 : 30 => 50 : 72 (72 - 50 = 22: too large)
88 : 55 : 33 => 55 : 75 (75 - 55 = 20: too large)

...and so forth. Notice the pattern: at each stage, the difference decreases by 2. Clearly, we'll need nine more decreases (a difference which is 18 smaller than 20) to get what we need. To get "88" for A, we multiplied the original ratio by 11. In nine more steps, we'll be multiplying by 20:

160 : 100 : 60 => 100 : 102 (102 - 100 = 2: just right!)

Hope that makes sense!