One number is 6 times a first number. A third number is 100 more than the first number. If the sum of the three numbers is 828​, find the numbers.

Answer 1

Givens:

1. One number is 6 times a first number
2. A third number is 100 more than the first number
3. Sum of the three numbers is 828
4. Find the numbers.

We will assign each number a variable and then form equations from what is given:

first number = x

second number = y

third number = z

1. 6x = y OR z (because this isn't clear in the question, there are two possible answers)
2. z = 100 + x
3. x + y + z = 828

Answer 1:

Assume 6x = y

x + 6x + (100 + x) = 828

8x + 100 = 828

8x = 728

x = 91

y = 6*x = 6*91 = 546

z = 100 + x = 100 + 91 = 191

x = 91, y = 546, z = 191 [Answer 1]

Answer 2:

Assume 6x = z

6x = 100 + x

5x = 100

x = 20

z = 100 + x = 100 + 20 = 120

x + y + z = 828

20 + y + 120 = 828

y = 688

x = 20, y = 688, z = 120 [Answer 2]

Answer 2

Write first number as x.

Write second number as 6x.

Write third number as x + 100.

From x + 6x + (x + 100), obtain 8x + 100, given as equal to 828.

8x + 100 − 100 = 828 − 100.

8x = 728 or x = 91.

6x = 546.

x + 100 = 91 + 100 or 191.

The three numbers are then (first number, second number, third number) = (91, 546, 191) and their sum equals 828.