Question

The sum of the tens digit and the hundreds digit of a number is three times the units digit. 1/5 of the sum of all three digits is 1 less than the units digit. Find all three-digit numbers that satisfy these conditions.

Answer 1

Answer: The numbers are:

965

875

785

695

Step-by-step explanation:

We have a 3 digit number that can be written as:

a*100 + b*10 + c

Where a, b, and c are single-digit numbers.

a is the hundreds

b is the tens

c is the unit.

We know that:

"The sum of the tens digit and the hundreds digit of a number is three times the units digit."

a + b = 3*c

" 1/5 of the sum of all three digits is 1 less than the units digit."

(a + b + c)/5 = c - 1

Then we have the two conditions:

a + b = 3*c

(a + b + c)/5 = c - 1

From the first one, we can write:

(a + b)/3 = c

Replacing that in the other equation we get:

(a + b + a/3 + b/3)/5 = a/3 + b/3 - 1

(4/3)*(a + b)/5 = (a/3 + b/3) - 1

(4/15)*(a + b) = (a + b)/3 - 1

(4/15)*(a + b) - (a + b)/3 = - 1

-(1/15)*(a + b) = -1

a  = 15 - b

Then we can give different values for b, and find the values of a and c, where a and c must be positive.

b = 9

a = 15 - 9 = 6

Then:

c = (9 + 6)/3 = 5

(notice that a + b = 15, then c will be always equal to 5, reggardless of the values of b and a).

This number will be:

695.

if b = 8, then:

a = 15 - 8 = 7

and c = 5, same as before.

the number is 785.

If b = 7, then:

a = 15 - 7 = 8

The number is 875

if b = 6, then:

a = 15 - 6 = 9

The number is:

965