The sum of the ages of mrs. spelman and mr. murphy are 78 years. six years from now mrs. spelman's age will be equal to 6 times ms murphy's age. how old is each now?

Step-by-step explanation:

There are no integer solutions to the problem as posed. (We suspect a typo, that the intention is 3 years from now, or that the ratio will be 5:1.)

Let m represent Murphy's age now. Then 78-m represents Spelman's age now, and the ratio in 6 years will be ...

  78 -m +6 = 6(m +6)

  84 -m = 6m +36 . . . . . collect terms

  48 = 7m . . . . . . . . . . . . add m-36

  48/7 = m = 6 6/7 . . . . . divide by the coefficient of m

Then Spelman's age now is ...

  78 -m = 78 -6 6/7 = 71 1/7

Spelman is 71 1/7; Murphy is 6 6/7.

If the ratio is 6:1 in 3 years, then Murphy is 9 and Spelman is 69.

If the ratio is 5:1 in 6 years, then Murphy is 9 and Spelman is 69.

Alternate solution method

I find it easiest to add 12 years to the total (each ages by 6 years), which will give a total age of 90 in 6 years. At that time, Murphy is 1/7 of the total of ages, so dividing that sum into parts with the appropriate ratio gives m'=90/7=12 6/7; s'=77 1/7. So, m=6 6/7; s=71 1/7.

Comment on the problem

We think there is a "typo" because the ratio of 6:1 means the future total should be a multiple of 7. Of course, 78+6 =84 is a multiple of 7, but adding 6 to the total will occur in 3 years, not 6 years.

because linear equations are a lot simpler than quadratics equations, and linear equations formula is a lot easier to memorize. that's why i think they're harder.