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three of four numbers have a sum of 22. if the average of the four numbers is 8, what is the fourth number

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5 Answers

Hi Heather.  I usually tell my students to solve these word problems by writing down EXACTLY what it says - "in math".  Your problems starts "3 of 4 numbers have a sum of 22".  So we'll write

a+b+c = 22.  (I simply chose to call the 4 #s a, b, c and d, and here I used 3 of them.)

Next bit of info in your problem is "the average of the 4 numbers is 8".  Well you know the definition of average - you take the sum of the numbers and divide by the number of numbers - so let's write that out exactly, "in math", for our problem:

 

(a + b + c + d) /4 = 8.

I hate fractions.  So let's simplify this by multiplying through by the 4 that's in the numerator on the left.  Or, you could think of that 8 on the right as 8/1, and cross multiply:

(a + b + c + d) = 32.

[One tidbit learned here to tuck away:  if you know the any two of average = sum/#, you know the 3rd.  Just rearrange the equation!  sum = (ave)(#), and # = (sum)/(ave) ]

 

We already know (a + b + c) = 22, so just substitute that in:

22 + d = 32

Now solve for d:

d = 32 - 22 = 10.

Laslty - check: (a + b + c + d) / 4 = 8?

(22 + 10) / 4 = 8?

32/4 = 8?

8 = 8 ? yes.  So the fourth number is 10.

The methodology to write down exactly what the problem tells you is a really good strategy to help you through many types of problems.

Good luck!

 

Hey Heather -- the pile of three is 22 ... the pile of four is 4x8 or 32 ...

the 4th # is the difference in piles ==> 10 

KEY CONCEPT: pile size is ave * (# of items) ... Best wishes, ma'am :)

Think about working this problem backwards.

First, you need to know how to find an average: Add up the numbers, and then divide that sum by the number of items you added. 

Since you know that the average is 8, and there are 4 items, just multiply those two numbers to get the sum they add up to, which is 32.

Because three of the numbers add up to 22, the missing number must be 10 by subtraction (32 minus 22).

Write the word problem as a math equation… horrible I know:

A + B + C = 22 Equation 1.

(A + B + C + D)/4 = 8 Equation 2.

Now substitute in Equation 1 into Equation 2. (A + B + C = 22)

(22 + D)/4 = 8

Now multiply both sides of the equation by the denominator 4:

4 X 22 + D = 8 X 4 (the 4’s cancel out in the fraction)

         4

22 + D = 8 x 4 or 32

Now subtract 22 from the left side and from the right!

D = 32 – 22 = 10 et voila!

Call the four numbers A, B, C, D.

"three of four numbers have a sum of 22" so that means A+B+C =22

"the average of the four numbers is 8" so that means (A+B+C+D) / 4 = 8

Now plug in what you know from the first equation into the second. (22 +D) /4 =8.

Now just solve using algebra.

22 +D/ 4 = 8. 4 is being divided on the left side...

 22+D =4 x 8 So that means it would be Multiplied if moved to the right side.

22+D =32 I want to move the 22. It's being added on the left

D=32-22=10. So just Subtract it on the right.