The sum of two numbers is 70. One number is smaller than 8 more than the other. What's the smallest number?

Question

A)33

b) 31

C)35

d) 45

Answer 1

There are two numbers.  We could represent them with the variables x and y, A and B, or S and L (for smaller and larger, if we knew which was which  -- but we don't).
 
"The sum of two numbers is 70"        means      x + y = 70
"One number is smaller than 8 more than the other"     means     x < 8 + y
"What's the smallest number?"   means we have to report the smaller number
 
           x < 8 + (70-x)          [substituting for y]
 
With an inequality, we may add the same value (positive or negative) to both sides and the truth of the inequality remains.  That means:
         2x < 78                 [adding x to both sides]
 
Also, with an inequality, we may multiply or divide both sides by the same positive amount and the truth of the inequality remains (note: with a negative value, it reverses).  That means:
          x < 39
 
But, is x the smaller number?  Well, look again at:     x + y = 70    and substitute for x:
         (70 - y) < 39
             -y < -31               (subtract 70 from both sides)
              y > 31                  (multiply by -1 reverses the inequality

 

This means that pairs of numbers that add up to 70 and for which x<39 and y>31 fit the requirements.  For example:

      x,   y

     30, 40     30+40=70 and 30<(40+8)

     20, 50     20+50=70 and 20<(50+8)

 

Which of the answers fits?

a) 33,  37      33+37=70  and  33<(37+8)     yes.

b) 31   39      31+39=70 and  31<(39+8)     yes

c) 35,  35      35+35=70 and 35<(35+8)    yes

d) 45,  25     45+25=70 and 45 is NOT the smaller number

 

Now, the problem said “smallest” rather that the “smaller” of the two numbers.  So, perhaps they meant, “What’s the smallest possible number?”  That would be 31.  And, 31 is smaller than 8 more than 39.

 

 

 

 

Answer 2

Let x = the smaller number.  The larger number is 8 more than the smaller one, or x+8.  Their sum is 70:

 

      x + (x+8) = 70

 

      2x + 8 = 70

 

      2x = 62

 

Can you finish it from here?