real world applications

Question

1. how many liters of a 30% solution must be added to how many litres of a 70% solution to make 50l of a 40% solution?
2. An investor invests $18,000 into two accounts, one that pays 1.5% simple interest, and the other that pays 2% simple interest. The investor earns $322.50 interest after one year. How much was invested in each account.?
3. A jigsaw puzzle factory has a cost of production C(x)=12.25+27,000 and revenue R(x)=20x. What is the break even-point? What is its cost and revenue at the break-even point?

Answer 1

mix 30% and 70% to get 50 Liters of 40%

let x =liters of 30%, y = liters of 70% = 50-x

.3x + .7(50-x) = .4(50)

.3x -.7x +35 = 20

.4x = 15

x = 15/.4 = 150/4 = 75/2 = 37.5 Liters of 30%

y = 50-37.5 = 12.5 Liters of 70%

that's 3 times as much 30% as 70%

30% is 3 times closer to 40% than 70% is

40-30 = 10

70-40=30

30 = 3 times 10

$18000 invested earning $322.50 with one investment at 1.5% and the 2nd at 2%

let x = amount invested at 1.5%, y=18,000- x = amount invested at 2%

.015x + .02(18000-x) = 322.5

(.015 - .020)x = 322.5 - 360

.005x = 37.5

x = 37.5/.005 = 37500/5 = $7,500 invested at 1.5%

y = $10,500 invested at 2%

322.5/18000 = about .01791 = about 1.8%

1.8-1.5 = .3

2-1.8 = .2

.3 is 3/2 times .2, amount invested at 2% is about 3/2 times amount invested at 1.5%

10,500/7500 = 105/75= 21/15 = 7/5= 1.4 which is close to 3/2=1.5, more or less

C(x) = 12.25x + 27K, R(x)= 20x

Profit = R-C = 7.75x - 27K = 0 at break even point

x = 27,000/7.75 =2700000/775= 108000/31= about 3484 rounded to nearest integer

Cost = Revenue at the break even point = zero profit point, C(x)=R(x)

Cost at break even point = 12.25(108,000/31) + 27,000

Revenue at break even point = 20(108,000/31) = same as C(108,000/31)= about $69,677 rounded to nearest dollar integer