Problem Solving

BTGModeratorVI wrote: ↑

Sun Aug 02, 2020 6:49 am

In a certain school, the ratio of boys to girls is 5 to 13. If there are 72 more girls than boys, how many boys are there?

A. 27
B. 36
C. 45
D. 72
E. 117


Answer: C
Source: Kaplan

One approach:

GIVEN: the ratio of boys to girls is 5 to 13
There are several possible cases that meet this condition:
- there are 5 boys and 13 girls
- there are 10 boys and 26 girls
- there are 15 boys and 39 girls
.
.
.
NOTE: We're also told that there are 72 more girls than boys.
So, as we continue listing possible cases, we'll keep track of the DIFFERENCE in the number of boys and girls
.
.
.
- there are 20 boys and 52 girls (there are 32 more girls than boys)
- there are 25 boys and 65 girls (there are 40 more girls than boys)
- there are 30 boys and 78 girls (there are 48 more girls than boys)
- there are 35 boys and 91 girls (there are 56 more girls than boys)
- there are 40 boys and 104 girls (there are 64 more girls than boys)
- there are 45 boys and 117 girls (there are 72 more girls than boys)

So, there must be 45 boys and 117 girls

Answer: C

BTGModeratorVI wrote: ↑

Sun Aug 02, 2020 6:49 am

In a certain school, the ratio of boys to girls is 5 to 13. If there are 72 more girls than boys, how many boys are there?

A. 27
B. 36
C. 45
D. 72
E. 117


Answer: C
Source: Kaplan

[spoiler=]This is a pretty simple problem, however I was looking for purely algebraic way(s) to solve it. For some reason have trouble expressing it properly, please help to set it up.[/spoiler][/quote]

Another approach:
Let B = # of boys
Let G = # of girls

The ratio of boys to girls is 5 to 13.
We can write: B/G = 5/13
Cross multiply to get: 13B = 5G

There are 72 more girls than boys
We can write G = B + 72
Alternatively, we can write: G - 72 = B

We now have two equations:
13B = 5G
G - 72 = B

Take bottom equation and multiply both sides by 5 to get: 5G - 360 = 5B
Now replace 5G with 13B [since the top equation tells us that 13B = 5G]
We get: 13B - 360 = 5B
Add 36 to both sides: 13B = 5B + 360
Subtract 5B from both sides: 8B = 360
Solve: B = 45

Answer: C

Cheers,
Brent