SOLUTION: The ratio of ages of 2 persons A and B is 3:5.The sum of their ages is 80. After 10 years, what will be the ratio of their age

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Question 229098: The ratio of ages of 2 persons A and B is 3:5.The sum of their ages is 80. After 10 years, what will be the ratio of their age
Answer by jsmallt9(3758) (Show Source):
You can put this solution on YOUR website!
If you often have trouble with age-related word problems, you may want to look at the lesson I wrote, "Tips for solving age-related word problems", on this site. I will be using these tips to solve your problem:

1. Write and expression for the age of each person at each point in time. Since we have two people, A and B, and two points in time, now and 10 years from now, we will write 4 expressions:
Let a = A's age now
Let b = B's age now
then
a+10 = A's age in ten years
b+10 = B's age in ten years.

2. Write equations using the relationships described in the problem. Write as many equations as you have variables in the expressions. Since I've used two variables above, I need two equations. (A one-variable solution will be shown below.) One relationship is the ratio:

The other relationship described is the sum:

3. Solve the equation (or system of equations). We have a system and there are many techniques for solving systems of equations. But most of them work only on systems of linear equations. As these equations are written, it does not look like they are linear (because of the b in the denominator in the first equation). (As it turns out the first equation actually is linear.) So we'll use the Substitution Method which begins by solving one equation for one of the variables. Solving the second equation for a:

Add negative b to (or subtract b from) each side:

Now we substitute into the other equation:

Solve this. We'll start by getting rid of the fractions by multiplying both sides by the Lowest Common Denominator (LCD). The LCD is 5b:

Add 5b to each side:

Divide both sides by 8:

Find the other variable using either equation:
a+b=80
a+(50)=80
a = 30
4. Answer the question. This last step is very important and often overlooked. It is easy to feel satisfaction with finding the solutions for a and b. But the question of the problem is "What is the ratio of their ages in ten years?". To answer this we'll be using our expressions (from step 1) for their ages in ten years. The ratio of their ages in ten years:

Substituting our solutions for a and b:

So the answer to the problem is: 2/3.

A one-variable solution. This is harder at the start because writing all four expressions with one variable can be tricky. But later it is easier because we will only need to write and solve one equation.
1. Write the expressions:
Let a = A's age now
Now how do we express B's age in terms of "a"? This can be the tricky part. From the sum relationship we could use:
Let 80-a = B's age now
(If we use the ratio relationship instead...
Let
3. Solve the equation. We'll multiply both sides by the LCD which is (80-a)(5)

4. Answer the question. The ratio of the ages in ten years: