What number must replace x
and y
for the three fractions to become equivalent?
6/14 = x/7 = 24/y
To find the equivalent fractions, we can cross-multiply:
6y = 14x
7x = 24y
To eliminate fractions, we can set y = 1 and solve for x:
6(1) = 14x
6 = 14x
x = 6/14 = 3/7
Now, let's set x = 1 and solve for y:
7(1) = 24y
7 = 24y
y = 7/24
Therefore, the numbers that should replace x and y for the three fractions to become equivalent are x = 3/7 and y = 7/24.
To find the values of x and y that would make the three fractions equivalent, we can set up a proportion and solve for x and y.
We have the following fractions: 6/14, x/7, and 24/y.
First, let's find the common denominator for these fractions, which is 14y since 14 and y are both denominators.
Now, we need to make each fraction have the same denominator of 14y.
For the first fraction, 6/14, we need to multiply both the numerator and denominator by (y/1). This gives us (6y)/(14y).
For the second fraction, x/7, the denominator is already 7, so we don't need to make any changes to it.
For the third fraction, 24/y, we need to multiply both the numerator and denominator by (14/1) to get (24 * 14)/(y * 1) = 336/y.
Now that we have all the fractions with the same denominator, we can set them equal to each other:
(6y)/(14y) = x/7 = 336/y
Since all three fractions are equal, we can set them equal to each other:
(6y)/(14y) = x/7 = 336/y
From here, we can solve for x and y.
First, let's solve for x. Cross-multiplying the first and second fractions gives us:
7 * (6y) = 14y * x
42y = 14xy
Dividing both sides by 14y gives us:
42 / (14y) = x
Simplifying:
3 / y = x
So, x = 3 / y.
Next, let's solve for y. Cross-multiplying the second and third fractions gives us:
7 * (336/y) = 14y * x
2352 / y = 14xy
Dividing both sides by 14x gives us:
(2352 / y) / (14x) = y
Simplifying:
(168 / y) / (3 / y) = y
168 / 3 = y^2
56 = y^2
Taking the square root of both sides:
y = ± √56
So, the values of x and y that would make the three fractions equivalent are:
x = 3 / y
y = ± √56