The difference of two numbers is 12. Two fifths of the greater number is six more than one-third of the lesser number. Find both numbers.
x - y = 12,
2x/5 = y/3 + 6,
Multiply both sides by 15, the common denominator:
6x = 5y + 90,
6x - 5y = 90,
Multiply both sides of 1st Eq by -6 and add the 2 Eqs:
x - y = 12
6x - 5y = 90
-6x + 6y = -72
6x - 5y = 90
y = 18,
Substitute 18 for y in the 1st Eq:
x - 18 = 12,
x = 30.
Solution set: x = 30, and y = 18.
Let's assume the two numbers are x and y, where x is the greater number and y is the lesser number.
According to the given information, we can set up two equations:
1) The difference of two numbers is 12:
x - y = 12
2) Two fifths of the greater number is six more than one-third of the lesser number:
(2/5)x = (1/3)y + 6
We can solve this system of equations using substitution or elimination. I will solve it using substitution:
From equation 1, we can express x in terms of y:
x = y + 12
Substitute this value of x in equation 2:
(2/5)(y + 12) = (1/3)y + 6
Now, we can solve for y:
Multiply both sides by 5 to eliminate the fraction:
2(y + 12) = 5((1/3)y + 6)
2y + 24 = (5/3)y + 30
Multiply both sides by 3 to eliminate the fraction:
6y + 72 = 5y + 90
Subtract 5y from both sides:
y + 72 = 90
Subtract 72 from both sides:
y = 18
Now, substitute the value of y back into equation 1 to find x:
x - 18 = 12
Add 18 to both sides:
x = 30
Therefore, the two numbers are 30 and 18.
To solve this problem, let's represent the two numbers with variables.
Let's call the greater number "x" and the lesser number "y".
We know that the difference between the two numbers is 12, so we can write the equation:
x - y = 12 (Equation 1)
We also have the information that two fifths of the greater number is six more than one-third of the lesser number. Let's translate this statement into an equation.
Two fifths of the greater number can be written as (2/5)x, and one-third of the lesser number can be written as (1/3)y.
According to the problem, (2/5)x is six more than (1/3)y, so we can write the equation:
(2/5)x = (1/3)y + 6 (Equation 2)
Now, we have a system of two equations (Equation 1 and Equation 2) that we can solve simultaneously to find the values of x and y.
To solve, we can use a method called substitution:
1. Solve Equation 1 for x:
x = y + 12
2. Substitute this expression for x in Equation 2:
(2/5)(y + 12) = (1/3)y + 6
3. Simplify and solve for y:
2(y + 12)/5 = (1/3)y + 6
Multiply both sides of the equation by the least common multiple of the denominators (5 and 3), which is 15 to eliminate the fractions:
6(y + 12) = 5(y) + 90
Simplify the equation:
6y + 72 = 5y + 90
4. Solve for y:
Subtract 5y from both sides and subtract 72 from both sides:
6y - 5y = 90 - 72
Simplify:
y = 18
5. Now that we have the value of y, we can substitute it back into Equation 1 to find x:
x = y + 12
x = 18 + 12
x = 30
Therefore, the two numbers are 30 and 18.