help pls! tried it a lot but just cant get it.?

Question

1. the difference of two numbers is 3 and the difference of their squares is 69 . find the numbers.
ans:13 and 10

2.two consecutive natural numbers are such that one-fourth of the smaller exceeds one-fith of the greater by 1. find the numbers.
ans:24 and 35

3.three consecutive whole numbers are such that if they be divided by 5, 3 ansd 4 respectively; the sum of the quotients is 40. find the numbers.
ans:50, 51 & 52

Answer 1

1. The first sentence gives you x - y = 3. We can change it to x = y + 3. The second equation is x^2 - y^2 = 69. Substitute the other equation for x.

(y + 3)^2 - y^2 = 69
Y^2 + 6y + 9 - y^2 = 69
6y + 9 = 69
6y = 60
y = 10
x = 13

2. Two consecutive natural numbers are x and x + 1.
1/4 x = 1/5(x + 1) + 1 Multiply by 20 to get rid of the fractions.
5x = 4(x + 1) + 20
5x = 4x + 4 + 20
5x = 4x + 24
x = 24
x + 1 = 25

3. Three consecutive numbers are x, x+ 1 and x + 2.

x/5 + (x + 1)/3 + (x + 2)/4 = 40
Multiply by 60 to get rid of the fractions.
12x + 20(x + 1) + 15(x + 2) = 2400
12x + 20x + 20 + 15x + 30 = 2400
47x + 50 = 2400
47x = 2350
x = 50
x + 1 = 51
x + 2 = 52

Answer 2

1. Let x and y be the numbers, with x being the larger. Then

x-y = 3, and x^2 - y^2 = 69. But x^2-y^2 = (x+y)(x-y), so you have

69 = (x+y)(x-y) = (x+y)3.

Now, use x-y=3 to obtain y=x-3, and substitute into the equation above:

69 = (x+x-3)3 = (2x-3)3 = 6x-9.

Move the -9 to the other side, and you get 78=6x, so x=13, and y=10.

2. Let x and y be the numbers, with x being the larger. Since they're consecutive, x=y+1. Also, y/4 = x/5 + 1. Substitute y+1 for x, and you get

y/4 = (y+1)/5 + 1.

Multiply both sides by 4:
y = 4(y+1)/5 + 4
Multiply both sides by 5:
5y = 4(y+1) + 20
Expand:
5y = 4y + 4 + 20
Solve for y, and you get y=24. Therefore x=25, not 35.

3. Let x,y,z be the numbers, in order. Then y=x+1, and z=y+1=x+2.

You have x/5 + y/3 + z/4 = 40, or
x/5 + (x+1)/3 + (x+2)/4 = 40.

Multiply both sides by 3*4*5 = 60, and you get
12x + 20(x+1) + 15(x+2) = 2400.
Multiply out:
12x + 20x + 20 + 15x + 30 = 2400.
Combine like terms:
47x = 2350.
Solve for x, and you get x=50, so y=51 and z=52.

Answer 3

i dont get it