NO!!! Word Problems????

Question

Linear Equations

1. An art dealer sold two artworks for $1520 thereby making a profit of 25% on the first work and 10% profit on the other, whereas if he had approached any exhibition he would have sold them together for $1535 with a profit of 10% on the first and 25% on the other artwork. Find the actual cost of each artwork.

2. A professor started a project with a group of students, 60% of whom were boys. Due to some unavoidable reasons, six girls couldn't turn up, so the professor had to make some changes in the group. He admitted six boys. In doing so, the percentage of boys in the project increased to 75%. Find the number of boys and girls involved in the project initially.

I am not good at word problems at all. My biggest weakness, any help would be appreciated. Thanks!

Answer 1

1)
actual $ of Art work #1 = x
actual $ of Art work #2 = y

In the first case...
A 25% profit on x = (1.25)x
A 10% profit on y = (1.10)y
so...
(1.25)x + (1.10)y = 1520

In the second case...
A 10% profit on x = (1.10)x
A 25% profit on y = (1.25)y
so...
(1.10)x + (1.25)y = 1535

Now you have two equations and two unknowns...
(1.25)x + (1.10)y = 1520
(1.10)x + (1.25)y = 1535

Solve one equation for one variable...
(1.25)x + (1.10)y = 1520
(1.25)x = 1520 - (1.10)y
x = 1216 - (0.88)y

Now put that in the other equation.
(1.10)x + (1.25)y = 1535
(1.10)(1216 - 0.88y) + 1.25y = 1535
1337.6 - 0.968y + 1.25y = 1535
0.282y = 197.4
y = 700

Now use that to find the other variable...
(1.25)x + (1.10)y = 1520
1.25x + (1.10)(700) = 1520
1.25x + 770 = 1520
1.25x = 750
x = 600

So, the art works were $600 and $700.
========================
2)
b = initial # of boys
T = total # of people in group

Originally, the boys were 60% of the group:
b = (0.60)T

He added 6 boys, so now the boys were 75% of the group:
b + 6 = (0.75)T

So the two equations are:
b = (0.60)T
b + 6 = (0.75)T

Substitute the first equation into the second...
(0.60)T + 6 = (0.75)T
6 = (0.75)T - (0.60)T
6 = (0.15)T
40 = T

There are 40 total students in the group.

b = (0.60)T = (0.60)(40) = 24 boys initially
40 - 24 = 16 girls initially

Answer 2

Let x = cost of 1st artwork
Let y = cost of 2nd artwork
Then 1.25 x + 1.1y = 1520, and
1.1x+1.25y = 1535
From the 1st equation, y = (1520-1.25x)/1.1
So put this in place of y in the 2nd equation getting:
1.1x +1.25(1520-1.25x)/1.1 =1535
1.21x +1900 - 1.5625x = 1688.5
-.3525x = -211.5
x = $600
1.1(600) +1.25y =1535
660 +1.25y = 1535
1.25y = 875
y = $700

x= # of boys originally
y = # of girls originally
x/(x+y) = .6
(x+6)/(x+y) = .75
x = .6x+.6y --> .4x =.6y --> x = 1.5 y
So (1.5y+6)/(1.5y+y) = .75
1.5y +6 = .75(2.5y) = 1.875y
.375 y = 6
y = 16 girls originally
x/(x+16)=.6
x = .6x + 9.6
.4x = 9.6
x = 24 boys originally