How to Combine Ratios

To combine two ratios, the shared letter must have the same part in both.

Here $b$ is already 3 in $a:b=1:3$ and in $b:c=3:4$, so no scaling is needed.

Keep each letter in its own place: $a=1$, $b=3$, $c=4$.

So $a:b:c=1:3:4$.

To combine two ratios, the shared letter must have the same part in both.

Here $b$ is already 7 in $a:b=4:7$ and in $b:c=7:2$, so no scaling is needed.

Keep each letter in its own place: $a=4$, $b=7$, $c=2$.

So $a:b:c=4:7:2$.

No number divides into all three parts, so $4:7:2$ is already in its simplest form.

The letter $b$ appears in both ratios, so scale each ratio until the two $b$ parts match. The LCM of 5 and 2 is 10.

Scale $a:b=3:5$ by 2 to get $6:10$. Scale $b:c=2:7$ by 5 to get $10:35$.

Now $b$ is 10 in both, so they can be joined: $a:b:c=6:10:35$. Remember that both parts of a ratio must be multiplied by the same number. You cannot simply write the two ratios side by side.

Green appears in both ratios, so make the green parts match. Green is 2 in red:green $=5:2$ and 4 in blue:green $=3:4$, and the LCM of 2 and 4 is 4.

Scale red:green $5:2$ by 2 to get $10:4$. Blue:green $3:4$ already has green as 4, so it stays as it is.

The question asks for red:green:blue, so keep that order: $10:4:3$.

$y$ appears in both ratios, so make the two $y$ parts match. The LCM of 6 and 4 is 12.

Scale $x:y=7:6$ by 2 to get $14:12$. Scale $y:z=4:5$ by 3 to get $12:15$.

Both parts of a ratio must be multiplied by the same number, so joining them gives $x:y:z=14:12:15$.

Bananas are 3 parts in both ratios, so the two ratios link straight away: apples:bananas:grapes $=5:3:2$.

The 5 parts of apples stand for 25 apples, so one part is $25÷5=5$.

Grapes are 2 parts, so there are $2\times 5=10$ grapes. (Bananas would be $3\times 5=15$, but the question asks about grapes.)