Math CentralQuandaries & Queries


Question from Ali, a parent:

Please help us. My daughter can't get this answer.
The sum of Louise's age and Todd's age is 34. 5 years ago the sum of twice Louise's age and 3 times Todd's age was 61. How old are they now?

We have two responses for you.

Hi Ali.

Let's use their names to represent their current ages.

a) Louise + Todd = 34.

Five years ago, Louise was (Louise - 5) years old and Todd was (Todd - 5) years old. At that time, twice her age was
2 x (Louise -5) and three times Todd's age was 3 x (Todd - 5). And the sum of these two was 61, so:

b) 2 x (Louise - 5) + 3 x (Todd - 5) = 61

This gives us two equations with two unknown quantities, so we can find out what those quantities are by solving each equation for one variable, then making them equal each other:

a) Louise + Todd = 34, so:
Louise = Todd - 34.

b) 2 x (Louise - 5) + 3 x (Todd - 5) = 61, so:
2 x (Louise - 5) = 61 - 3 x (Todd - 5)
Louise - 5 = [ 61 - 3 x (Todd - 5) ] / 2
Louise = 5 + [ 61 - 3 x (Todd - 5) ] / 2

If two different expressions equal the same third expression (Louise in this case), then they equal each other, so:
Todd - 34 = 5 + [ 61 - 3 x (Todd - 5) ] / 2

Now you just have one unknown value to find. Solve for Todd's age anduse it to find Louise's age.

Hope this helps,
Stephen La Rocque.

Hi Ali,

You can use algebra to help solve this problem. First, we translate the English words into algebraic expressions. To do this, ask yourself what numbers are you trying to find: Louise's current age, and Todd's current age. Next, translate these values into algebraic expressions by simply choosing a symbol to represent each. Here are the symbols I chose:

L = Louise's current age
T = Todd's current age

Okay. So far so good. In order to figure out the values of L and T, we must know something about how they are related to each other. For example, if I knew L was twice as much as T, and I knew T was 4, then I could figure that T is 4 and L is 8. This is a simpler problem than the one you have, but the idea is the same.

So, look back to the question and see what you know about T and L and how they are related. You are told that "the sum of Louise's age and Todd's age is 34" and "5 years ago the sum of twice Louise's age and 3 times Todd's age was 61". I'll help translate the second part into an algebraic relation, and let you try the first part.

"5 years ago the sum of twice Louise's age and 3 times Todd's age was 61":

Okay, 5 years ago Louise's age was (L-5), and Todd's age was (T-5) [Does this make sense to you?] . So the above statement can be written algebraically as:

2*(L-5) + 3*(T-5) = 61, which can be simplified:
2L -10 + 3T-15 = 61
2L +3T = 61 + 10 + 15
2L + 3T = 86

I will leave it to you to translate "the sum of Louise's age and Todd's age is 34" into an algebraic expression. Once you do, you will have two equations and two unknown values. We have answered many questions about how to do this. You can search our database for "simultaneous equations" or "linear equations" or "two unknowns". One example is:


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