The ratio of the age of a man and his wife is 4:3. After 4 years, this

Solution:

Let 4x and 3x be the current age of man and his wife

• After 4 years
• \(\frac{(4x+4)}{(3x+4)} = \frac{9}{7}\)

o \(28x + 28 = 27x + 36\)
o \(x = 8\)
• The current age of man = \(4*8 = 32\)
• The current age of man = \(3*8 = 24\)

Assume that the couple got married y years ago.

• \(\frac{(32-y)}{(24-y)} = \frac{5}{3}\)

o \(96 -3y = 120 -5x\)
o \(2y =24\)
o \(y = 12\).

Hence, the correct answer is Option B.

given
m/w=4/3
m=4w/3
m+4/w+4= 9/7
7m+28=9w+36
w=24
and m= 32
so
now
32+x/24+x = 5/3
solve for x
x= 12
IMO B

Given: The ratio of the age of a man and his wife is 4:3. After 4 years, this ratio will be 9:7.

Asked: If at the time of the marriage, the ratio was 5:3, then how many years ago they got married?

Let the age of man and wife be m & w respectively.
and let the number of years before they got married = t years

\(\frac{m}{w} = \frac{4}{3}\) (1)
\(\frac{m+4}{w+4} = \frac{9}{7}\) (2)
\(\frac{m-t}{w-t }= \frac{5}{3}\) (3)
t = ?

\(\frac{m}{m-w} = 4\)
\(\frac{m+4}{m-w} = \frac{9}{2}\)
\(\frac{m-t}{m-w} = \frac{5}{2}\)

\(\frac{m}{m+4} = \frac{8}{9} = \frac{32}{36}\)
m = 32
w = 24

\(\frac{m}{m-t} = \frac{8}{5}\)
\(\frac{t}{m} = \frac{3}{8}\)
\(t = \frac{3}{8} * 32 = 12 \)

IMO B

Ideally, I would have liked to plug in the answer choices. But, because it asks how many years ago they got married, I can't actually do it, since I don't have any actual ages or dates to plug in. We're stuck with algebra

So, let's draw a little timeline to keep everything organized...

marriage: husband = 5x, wife = 3x

now: husband = 4y, wife = 3y

4 years from now: husband = 9z, wife = 7z

We can use the fact that 4 years will pass to set up two equations and solve for y and z:

4y + 4 = 9z (man's age)
3y + 4 = 7z (woman's age)

Multiply the top equation by 3 and the bottom by 4:

12y + 12 = 27z
12y + 16 = 28z

z = 4

Plug back in:

4y + 4 = 9(4)
4y + 4 = 36
4y = 32
y = 8

That lets us fill in part of the table!

marriage: husband = 5x, wife = 3x

now: husband = 4y = 4(8) = 32, wife = 3y = 3(8) = 24

4 years from now: husband = 9z = 9(4) = 36, wife = 7z = 7(4) = 28

So, how long ago did they get married?

At this point, I'm going to switch to testing the answer choices, to avoid having to do even more algebra. If the husband and wife are now 32 and 24, how many years ago was their ratio 5:3?

(a) 8 years ago, the husband was 24 and the wife was 16, for a ratio of 3:2.
(b) 12 years ago, the husband was 20 and the wife was 12, for a ratio of 5:3. B is the correct answer (Although it's a good lesson about checking your answers when you write a problem, since this one definitely implies that a 20 year old married a 12 year old... )