Question 1065231: Sally is 3 years older than Helen. The product of their ages is 70. How old is Sally?
Found 3 solutions by Alan3354, KMST, ikleyn:
Answer by Alan3354(69443)   (Show Source):
Answer by KMST(5328)  (Show Source):
You can put this solution on YOUR website! THE TIP:
To find pairs of whole-number factors of a number,
you can look to the prime factorization,
such as 70=2×5×7
and work from there.
However, sometimes, it is easier to try possible factors,
starting with 1 and going up.
For example,
70=1×70
1 is always a factor,
but 2 also divides 70 evenly,
so we find that
70=2×35 .
We realize that 3, and 4 are not factors of 70,
but 5 is, and we find that
70=5×14 .
We know that 6=2×3 is not a factor of 70,
because we had figured out before that 3 is not a factor.
Next comes 7, and we know that
70=7×10.
The 4 pairs of factors listed above are all there is.
I know that because, for a pair of factors,
either both factors are the same,
as in 8×8=64 and 9×9=81,
or one factor is smaller,
and the other is larger.
Since 70 is between 64 and 81,
its pairs of factors will have
one factor smaller than 9,
and another one larger.
Since 4 is not a factor,
8=2×4 I
cannot be the smaller of a pair of factors,
so I stop looking for factor pairs after
7×10=70.
THE ALTERNATE SOLUTION:
It is possible that your teacher expected you to set up and solve a quadratic equation. One way to do that:
 = Helen's age (defining a variable)
So,  = Sally's age.
Since the product of their ages is 70,
 is our equation.
 <--->  <---> 
Applying the quadratic formula to that last equation,
The positive solution is 
The negative solution,  ,
does not make sense as Helen's age.
So, Helen is  , and Sally is  .
Answer by ikleyn(52792)  (Show Source):
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