Solving and Reasoning Problems

Question

Linda is planning to give her money to two friends. She will give her friend Marcia $10 more than she gives her friend Karen. The ratio of the two amounts she gives away is 5 to 3. How much money does she give away.

Answer 1

Yaris, word problems generally involve easy math preceded by confusing words.

The first thing to do is identify the quantities involved, and name them.

Let m be the amount she gives to Marcia.

Let k be the amount she gives to Karen.

Next translate the other words into equations.

Translate “she gives Marcia” into “m”

translate “$10 more than she gives Karen” into “= 10 + k”

So far, we have one equation

m = 10 + k

But there’s more:

Translate “the ratio of the two amounts is 5 to 3” into “m / k = 5 / 3”

You may wonder how I know that it’s m/k rather than k/m. Since 5 is greater than 3, we want the larger value in the numerator. Since m is greater than k, the numerator should be m.

So now you have two equations in two unknowns

m = 10 + k

m / k = 5 / 3

We assume you know how to solve these equations, Yaris, but here’s a hint for the folks who haven’t gotten there yet.

substitute (10 + k) for m into the other equation

(10 + k) / k = 5 / 3

now put over a common denominator by multiplying the left hand side by 3/3 and the right hand side by k / k

3 * (10+ k) / 3k = 5k / 3k

Next, remove the 3k denominator

3 * (10+ k) = 5k

At this point you can certainly solve for k.

Then add 10 to get m

and you’re done.

Answer 2

Let's start by declaring the variables. M will stand for the amount Linda will give to Marcia, and K will stand for the amount Linda gives to Karen. Now we just convert English to math.

M = K + 10

M/K = 5/3 Since we know from above that Marcia's amount is larger!

Solving the second equation for M we get, M = (5*K)/3

Let's substitute this into the first equation and solve for K. We get K = 15.

And using this value for K in the first equation, we get M = 25.

Now let's check the answer. using the second equation, we get 25/15 = 5/3. If we cross multiply, we get 5*15 = 3*25. This is true!

So just to be complete, Linda plans to give Marcia $25 and Karen $15.

Hope this helps!