One number exceeds another by 4. If three times the larger number is divided by the smaller number, the quotient is 4 and the remainder is 5. Find the numbers.
(I am trying to figure out how to explain how to do this to my son. He is struggling and will be going to tutoring, but that won't help me explain it for his homework due tomorrow)
smaller number --- x
larger number ---- x+4
3(x+4)/x = 4 + 5/x
times x
3(x+4) = 4x + 5
3x + 12 = 4x + 5
7 = x
the smaller is 7 , the larger is 11
check:
33/7 = 4 with remainder 5
OK, all is good
To help your son understand and solve this problem, you can follow these steps:
Step 1: Let's represent the smaller number as 'x' and the larger number as 'y'.
Step 2: Since "One number exceeds another by 4," we can write the equation as y = x + 4.
Step 3: According to the second statement, "Three times the larger number is divided by the smaller number, the quotient is 4 and the remainder is 5." This can be represented as (3y) / x = 4 + 5/x.
Step 4: Simplify the equation in step 3:
(3y) / x = (4x + 5) / x
Step 5: Cross multiply to eliminate the fraction:
3y * x = 4x + 5
Step 6: Distribute and simplify:
3xy = 4x + 5
Step 7: We have two equations now:
y = x + 4 (from step 2)
3xy = 4x + 5 (from step 6)
Step 8: Substitute the value of y from step 2 into the second equation:
3x(x + 4) = 4x + 5
Step 9: Expand and simplify:
3x^2 + 12x = 4x + 5
Step 10: Rearrange the equation to set it equal to zero:
3x^2 + 12x - 4x - 5 = 0
Step 11: Combine like terms:
3x^2 + 8x - 5 = 0
Step 12: Solve the quadratic equation using factoring, completing the square, or the quadratic formula. In this case, the quadratic equation doesn't factor easily, so we will use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
Step 13: Substitute the values into the quadratic formula:
x = (-(8) ± sqrt((8)^2 - 4(3)(-5))) / 2(3)
Step 14: Simplify the quadratic formula expression
x = (-8 ± sqrt(64 + 60)) / 6
x = (-8 ± sqrt(124)) / 6
Step 15: Calculate the value of x using a calculator or a math tool. It will yield two solutions, x1 and x2.
Step 16: Substitute the values of x1 and x2 into the equation in step 7 to find the values of y1 and y2.
Step 17: The solutions (x, y) are the pairs (x1, y1) and (x2, y2).
By following these steps, you will be able to find the numerical values for the smaller and larger numbers, meeting the requirements of the problem.