Two fractions have the same denominator,15. The difference between the fractions is 2/5,and the result of dividing one of them by the other is 4/7. What are the fractions?
x/15 - y/15 = 2/5 = 6/15
so
x - y = 6
y/x = 4/7
so
y = 4 x/7
x - 4x/7 = 6
3 x/7 = 6
x = 14
y = 8
so
14/15 and 8/15
I don't understand
To solve this problem, let's assign variables to the fractions. Let's call the fractions a/b and c/b, where a/b is the larger fraction.
Given that the difference between the fractions is 2/5, we can set up the equation:
a/b - c/b = 2/5
Since both fractions have the same denominator, we can combine them:
(a - c)/b = 2/5
Next, we are told that the result of dividing one fraction by the other is 4/7. Using the information, we can set up another equation:
(a/b) / (c/b) = 4/7
To simplify the equation, we can multiply both sides by (b/b) to cancel out the denominators:
a/c = 4/7
Now we have a system of equations:
1. (a - c)/b = 2/5
2. a/c = 4/7
To find the values of a and c, we can solve this system of equations.
First, let's solve equation 1:
Using cross-multiplication, we can rewrite equation 1 as (a - c) * 5 = 2 * b.
This simplifies to 5a - 5c = 2b.
Next, let's solve equation 2:
Using cross-multiplication, we can rewrite equation 2 as a * 7 = c * 4.
This simplifies to 7a = 4c.
Now we have a system of equations:
1. 5a - 5c = 2b
2. 7a = 4c
To simplify this system of equations, we can divide both sides of equation 2 by 7:
a = (4/7) * c
Substituting this into equation 1:
5((4/7)c) - 5c = 2b
(20/7)c - 5c = 2b
(20c - 35c)/7 = 2b
-15c/7 = 2b
To equate the denominators on both sides, we can multiply both sides of the equation by 7:
-15c = 14b
Now we have a new equation:
3. -15c = 14b
Since we have three equations, we can solve for a, b, and c by solving the system of equations 2 and 3 simultaneously.
First, let's solve equation 2 and 3:
Multiply equation 3 by 4 to eliminate the fraction:
-60c = 56b
Now we have two equations:
2. 7a = 4c
3. -60c = 56b
We can solve for c by setting equation 2 equal to equation 3:
7a = (-15/14)(-60c)
7a = 900c/14
a = (900/14) * (1/7)c
a = 900c/98
a = 450c/49
Now we have found a relationship between a and c. Let's substitute this relationship back into equation 2 to solve for b:
7(450c/49) = 4c
3500c = 392b
b = (3500/392)c
b = 625c/70
b = 125c/14
Now we have expressions for a, b, and c:
a = 450c/49
b = 125c/14
c = c
Since we know that the fractions have the same denominator of 15, we can set 125c/14 = 15:
125c = 210
c = 210/125
c = 42/25
Now we can substitute this value of c back into the expressions for a and b:
a = (450 * (42/25)) / 49
a = 1890/49
a = 210/7
b = (125 * (42/25)) / 14
b = 1050/70
b = 15/1
Therefore, the fractions are 210/7 and 15/1.