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Question 1205514: The larger of two positive numbers is 5 more than the smaller. If their product is
84, what are the numbers?
Found 4 solutions by MathLover1, ikleyn, greenestamps, Edwin McCravy:
Answer by MathLover1(20850)   (Show Source):
Answer by ikleyn(52908)  (Show Source):
You can put this solution on YOUR website! .
Let x be the number exactly half-way between the two unknown positive integer numbers.
Since the difference between the numbers is 5, it is clear that the numbers are (x-2.5) and (x+2.5).
Then their product is (x-2.5)*(x+2.5) = x^2 - 2.5^2 = x^2 - 6.25.
It gives us this equation for x
x^2 - 6.25 = 84.
From this equation
x^2 = 84 + 6.25 = 90.25.
From this point, we find mentally x =  = 9.5.
Then the lesser of two integer positive numbers is 9.5-2.5 = 7,
while the greater of these two numbers is 9.5+2.5 = 12.
ANSWER. The numbers are 7 and 12.
Solved (mentally).
Answer by greenestamps(13215)  (Show Source):
You can put this solution on YOUR website!
Since the problem is easily solved quickly by finding two positive integers whose difference is 5 and whose product is 84, it is clear that a formal algebraic solution was desired.
You can set the problem up using two variables, as one tutor did; but that seems like a long path to the solution. The problem is easily set up using a single variable.
With the standard elementary algebraic solution, the smaller number is x and the larger is x+5; then
x(x+5)=84
x^2+5x=84
x^2+5x-84=0
To solve this by factoring, we need to find two numbers whose product is 84 and whose difference is 5 -- which is exactly what the original problem required us to do. So the standard formal algebraic solution method doesn't help us much. But, continuing....
(x+12)(x-7)=0
x=-12 or x=7
The problem requires positive numbers for the solution, so the smaller number is x=7 and the larger is x+5=12.
ANSWERS: 7 and 12
To solve the problem using a formal algebraic method that DOES get you to the answer, use the common "trick" shown by the other tutor.
Since the numbers differ by 5, let x be the number halfway between the two numbers you are looking for. Then the two numbers are x+2.5 and x-2.5; and solving the equation that says the product of the two numbers is 84 is easy:
(x-2.5)(x+2.5)=84
x^2-6.25=84
x^2=90.25
x=sqrt(90.25)=9.5
Then the answers are x-2.5=7 and x+2.5=12.
That's a useful "trick" you can learn to use for setting up a large number of different kinds of problems where part of the given information is the difference between two numbers.
Answer by Edwin McCravy(20065)  (Show Source):
You can put this solution on YOUR website!
I've often wondered about problems like this:
L = S+5
L*S = 84
(S+5)*S = 84
S2+5S = 84
S2+5S-84 = 0
To factor that, you have to think of two integers that have
product 84 and difference 5, which is exactly what the problem
asks for in the first place!! So you have to essentially solve
the problem in your head in order to solve it by factoring!!
That seems so silly!
I guess to keep it from being silly, you'd have to complete the
square or use the quadratic formula, but that's much harder than
thinking up 7 x 12 = 84 and 12 - 7 = 5.
When I was teaching, I never assigned this type of problem,
especially when teaching factoring binomials, but every textbook
would always include them!
Edwin
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