Word problem

Question

A university professor received a government grant for M dollars to be spent on research.  25 percent of the grant was appropriated for indirect costs, so the professor used 20 percent of what was left to buy equipment.  What percent of the original sum of M dollars was either given to the university or spent on equipment

Answer 1

Assuming that after 25% of the grant was appropriated for indirect costs, and 20% of what was left to buy equipment, the remainder was given to the university:

 

M = the original amount of grant money

Let X = indirect costs

Let Y = equipment costs

Let Z = amount of $ given to university

 

25% of the grant was appropriated for indirect costs:

X = 0.25M

 

20% of what was left was used to buy equipment:

what was left = M-X

Y = 0.2(M-X) = 0.2(M-0.25M) = 0.2(0.75M) = 0.15M

 

Money given to university is 80% of what was left after indirect costs (since remaining 20% used to buy equipment)

Z = 0.8(M-X) = 0.8(M-0.25M) = 0.8(0.75M) = 0.6M

 

We can check this by noting that X + Y + Z should add up to M

X = 0.25M, Y = 0.15M, Z = 0.6M

0.25M + 0.15M + 0.6M = M 

 

Percent of original sum of M dollars given to university = 60%

Percent of original sum of M dollars spent on equipment = 15%

Alternate

Another way to check/solve this problem is to start with an assumed starting value of M

i.e.:

Let M = $100

25% of M = indirect costs = 0.25($100) = $25

 

What is left = $75

20% of $75 = equipment costs = 0.2($75) = $15

 

Given to university = M - indirect costs - equipment costs = $100 - $25 - $15 = $60

 

"What percent of original sum of M dollars was given to university?" translates into

(x%)$100 = $60 → x = 60%

 

"What percent of original sum of M dollars was spent on equipment?" translates into
(x%)$100 = $15 → x = 15%