The sum of three numbers is 61. The difference of the largest and smallest is 47, and the sum of the two smallest is 18. Find the numbers

Question

No additions. Struggling to find how to layout the equations to solve

Answer 1

Let the three numbers be x,y,z  (from smallest to largest)

 

Translate:

  "The sum of three numbers is 61"    means    x+y+z=61   [eq1]

  "The difference of the largest and smallest is 47"    means   z-x=47    [eq2]

  "the sum of the two smallest is 18"   means x+y=18    [eq3]

 

   x+y+z = 61          [eq1]

   x+y     = 18          [eq3]

------------------   [elimination; subtract equations]

           z = 43

  -x    +z = 47    [from eq2]

 ----------------  [elimination; subtract equations]

   x         = -4

 

 x + y + z = 61   [eq1]

-4 + y + 43 = 61      [substitute values for x and z]

       y = 22              [subtract 39 from both sides]

 

Check:

  Is   -4 + 22 + 43 = 61  ?

        61 = 61   ?yes

  Is 43 - (-4) = 47   ?

          47 = 47   ?yes

  Is  -4 + 22 = 18   ?

           18 = 18   ?yes

 

Answer 2

Hi Gina,

 

When you get distinct pieces of information like this, and it's hard to see exactly how they relate to one another, a system of simultaneous equations can be very helpful. Here's how I would coach a student through a solution:

 

 

 

PART I: Find the math in this this word problem.

 

If we name the three numbers A (the smallest), B, and C (the largest), then the three pieces of information provided can be framed in three equations (one each):

• "The sum of three numbers is 61" becomes:                             A + B + C = 61      (Equation 1)
• "The difference of the largest and smallest is 47" becomes:        C - A = 47             (Equation 2)
• "The sum of the two smallest is 18" becomes:                           A + B = 18            (Equation 3)

 

To solve a system of simultaneous equations, you need at least as many equations as unknowns. We have three equations and three unknowns, above. Perfect!

 

 

 

PART II: Solve the system with substitution.

 

• Equation 3 tells us that any time the term "A + B" appears, it can be replaced by 18; they're equivalent.
• So, we can solve for C in Equation 1 by plugging 18 in for A + B:
• A + B + C = 61   (Equation 1)
• 18 + C = 61        (substitute 18 for A + B, thanks to Equation 3)
• C = 61 - 18         (subtract 18 from both sides to get C alone)
• C = 43                (simplify)
• Now that we know C, the largest number, we can solve for A using Equation 2:
• C - A = 47           (Equation 2)
• 43 - A = 47         (substitute 43 in for C)
• -A = 47 - 43        (subtract 43 from both sides to get A alone)
• A = -47 + 43       (multiply both sides by -1 so that we're solving for regular A, not negative A)
• If you don't like working with negative numbers, you can re-write this as A = 43 - 47
• A = -4                 (simplify)
• Now that we know A and C, we can solve for B by plugging the values of A and C into Equation 1 or Equation 3. Either one will work. Here's a solution plugging A and C into Equation 1:
• A + B + C = 61       (Equation 1)
• -4 + B + 43 = 61     (substitute -4 in for A and 43 in for C)
• B + 43 = 61 + 4      (add 4 to both sides)
• B = 61 + 4 - 43       (subtract 43 from both sides to get B alone)
• B = 22                    (simplify)
• Let's check our answer by solving for B with Equation 3, as well:
• A + B = 18              (Equation 3)
• -4 + B = 18             (plug -4 in for A)
• B = 18 + 4              (add 4 to both sides to get B alone)
• B = 22                    (simplify)
• They match! So, we should be good to go. If you're worried, you can plug the values found for A, B, and C back into Equation 1 to verify that they sum to 61.

 

Hope this helps.

Take care,

Caitrin

Answer 3

Hi Gina,

 

Here's how you solve this:

 

a= largest

c= middle

b= smallest

 

1) a + b + c = 61

2) a - c = 47

3) b + c = 18

 

 

Lets start with the first and third equations. 

We know b + c = 18, so we can plug this into the first equation.

 

a + b + c = 61

a + 18 = 61

a = 43

 

Now we can move onto equation two.

 

a - c = 47

43 - c = 47

-c = 4

c = -4

 

We can put these back into the first equation to find the third number.

 

43 + b - 4 = 61

b + 39 = 61

b = 22

 

Your three numbers are 43, 22, and -4

 

 

There are multiple ways you can use the three equations to check your work!

 

Hope this helps,

Emily