How many divisions are there? Show solution

Question

240 reams of bond paper were to be used in printing to be divided equally among a certain number of divisions. If there had been four divisions more, each would have received 4/5 as many. How many divisions are there? Show solution. Another question. A number of financial institutions agreed to contribute equal amount of certain development project costing 6 million. If two more institutions would have been contributed equally, it would have cost each institution $0.50 million less. How many institutions contributed to the project? Show solution

Answer 1

If the number of divisions is d, then each division gets 240/d reams. If there were d+4, then 240/(d+4)=⅘(240/d).

So 1/(d+4)=4/5d. 5d=4(d+4), 5d=4d+16, d=16.

CHECK

240/16=15 reams for each of 16 divisions. ⅘ of 15=12 reams for each of 20 divisions.

Note that the number of divisions is independent of the number of reams of paper. The same answer would apply if there has been 2000 reams: 16×125=20×100=2000, 100/125=⅘.

If f is the number of financial institutions, 6/f million dollars is the individual contribution. For f+2 institutions the individual contribution is 6/f-0.5 million dollars.

Therefore, f(6/f)=(f+2)(6/f-0.5), 6=6-0.5f+12/f-1, 0=-0.5f+12/f-1, 0=-0.5f²+12-f, 0=f²+2f-24=(f+6)(f-4).

Since f cannot be negative, f=4.

CHECK

Each institution contributes 6/4=3/2=$1.5m (4 institutions). If there were 6 institutions their individual contribution would be $1m. 4×1.5=6×1=$6m.