Find two numbers whose sum is 90 and one-third of the smaller number is equal to one seventh of the larger number
Let the numbers be x and y, with x the smaller number.
x + y = 90
x/3 = y/7
7x = 3y
x = (3/7) y
Now substitute that x in the first equation.
(10/7)y = 90
y/7 = 9
y = 63
x = 27
Let's assume the smaller number as "x" and the larger number as "y".
According to the given information:
1) The sum of the two numbers is 90:
x + y = 90
2) One-third of the smaller number is equal to one-seventh of the larger number:
(1/3)x = (1/7)y
To solve these two equations, we can use the method of substitution:
Step 1: Solve the second equation for x:
(1/3)x = (1/7)y
Multiply both sides by 3:
x = (3/7)y
Step 2: Substitute the value of x from the second equation into the first equation:
(3/7)y + y = 90
Multiply both sides by 7 to eliminate the fraction:
3y + 7y = 630
Combine like terms:
10y = 630
Divide both sides by 10:
y = 63
Step 3: Substitute the value of y back into the second equation to find x:
x = (3/7)y
x = (3/7) * 63
x = 27
Therefore, the two numbers are 27 and 63, as the smaller number is 27 and the larger number is 63, which satisfy the given conditions.
To find two numbers that satisfy these conditions, we can use a system of linear equations.
Let's assume the smaller number is x and the larger number is y.
According to the problem statement, we have two conditions:
1) The sum of the two numbers is 90: x + y = 90.
2) One-third of the smaller number is equal to one-seventh of the larger number: (1/3)x = (1/7)y.
To solve this system of equations, we can multiply both sides of the second equation by 7 to eliminate the fractions. This will give us:
7(1/3)x = 7(1/7)y,
(7/3)x = (1/7)y.
Now, we can rewrite the equation as:
(7/3)x - (1/7)y = 0.
To eliminate the fractions, we can multiply both sides of this equation by 3 * 7 = 21:
21(7/3)x - 21(1/7)y = 0,
49x - 3y = 0.
Now we have a system of equations:
x + y = 90,
49x - 3y = 0.
We can solve this system of equations using any method like substitution, elimination, or graphing. Let's use the elimination method to solve it:
Multiply the first equation by 3:
3x + 3y = 270.
Now, add this equation to the second equation to eliminate y:
(49x - 3y) + (3x + 3y) = 0 + 270,
52x = 270,
x = 5.19 (rounded to two decimal places).
We can substitute this value of x into the first equation to find y:
(5.19) + y = 90,
y = 90 - 5.19 = 84.81 (rounded to two decimal places).
Therefore, the two numbers that satisfy the given conditions are approximately x = 5.19 and y = 84.81.