How many pennies were in the jar at the start?

Question

I need help finding the equation for this problem!!

 

Alice takes one-third of the pennies from a large jar. Then Ben takes one-third of the remaining pennies from the jar. Finally, Carla takes one-third of the remaining pennies from the jar leaving 40 pennies in the jar. How many pennies were in the jar at the start?

Answer 1

You may solve this problem in parts using arithmetic or you may develop an algebra formula to solve it.

 

Let's do the arithmetic first:
  "Alice takes one-third of the pennies"   that leaves two-thirds of the pennies in the jar
 
  "Ben takes one-third of the remaining pennies"   so Ben takes one-third of two-thirds.
                "of" usually means multiply    (1/3)(2/3)=2/9     Ben takes 2/9th of the pennies.
                 that leaves    (2/3) - (2/9)  in the jar
                                       (6/9) - (2/9)
                                             (4/9)
 
  "Carla takes one-third of the remaining pennies"     means Carla takes  (1/3)*(4/9)=(4/27)th
                the number of pennies left is
                                   (2/3) - (2/9) - (4/27)
                                      (4/9) - (4/27)
                                        (12/27) - (4/27)
                                             8/27 of the original number
 
If 40 is 8/27 th of the original number, what is the original number?
     well 1/8th of 40 is 5 and 1/8th of 8/27 is 1/27,  so 1/27th of the original number is 5
           the original number must be 27*5 = 135
 
 

Now, let's use algebra.  Let x=original number of pennies in the jar

 

  "Alice takes one-third of the pennies"   Alice takes (1/3)x; that leaves (2/3)x in the jar

 

  "Ben takes one-third of the remaining pennies"   so Ben takes one-third of two-thirds of x.

                "of" usually means multiply    (1/3)(2/3)x=(2/9)x     Ben takes 2/9th of the pennies

                 that leaves    (2/3)x - (2/9)x  in the jar

                                       (6/9)x - (2/9)x

                                             (4/9)x

 

  "Carla takes one-third of the remaining pennies"     means Carla takes  (1/3)*(4/9)x=(4/27)(x)

                the number of pennies left is

                                   (2/3)x - (2/9)x - (4/27)x

                                      (4/9)x - (4/27)x

                                        (12/27)x - (4/27)x

                                             (8/27)x

 

If 40 is 8/27 th of the original number, what is the original number?

      40 = (8/27)x                        [remember "of" means multiply]

       (27/8)(40) = x                        [multiply each side by (27/40)

            135 = x                     [there were originally 135 pennies] 

Answer 2

Hi Mimma, good question and takes some thought about how to solve it.

 

Let x represent the total number of pennies in the jar before anyone takes any out.

 

Step 1: Alice takes 1/3 of the pennies, or 1/3 * x.

 

That leaves how many pennies still in the jar? Answer: 2/3 * x, taking out 1/3 of the pennies leaves 2/3 still in the jar.

 

Step 2: Ben takes 1/3 of the remaining pennies from the jar, or 1/3 * (2/3 * x).

 

This is 2/9 * x. That leaves how many pennies still in the jar? Answer: 4/9 * x, because before Ben took his 1/3 share of the pennies, there were 2/3 or 6/9 * x pennies still in the jar. 6/9 - 2/9 leaves 4/9 * x pennies still in the jar.

 

Step 3: Carla takes 1/3 of the remaining pennies from the jar, or 1/3 * (4/9 * x).

 

This is 4/27 * x pennies.

 

Step 4: Set up the equation to solve for x.

 

x - (1/3 * x) - (2/9 * x)  - (4/27 * x) = 40

 

We can remove the x from all four terms on the left giving:

 

x * (1 - 1/3 - 2/9 - 4/27) = 40

 

The numbers in the parenthesis have a common denominator of 27.

 

1 = 27/27, 1/3 = 9/29, 2/9 = 6/27, 4/27 = 4/27

 

This gives: x * (27/27 - 9/27 - 6/27 - 4/27) = 40

 

x * (8/27) = 40

 

Multiply both sides by 27 to remove the fraction gives:

 

x * 8 = 40 * 27 = 1080

 

Divide both sides by 8 gives:

 

x = 135.

 

Solution: the penny jay started with 135 pennies.

 

Check:

 

substitute 135 for x in the step 4 equation gives:

 

135 - (1/3 * 135) - (2/9 * 135) - (4/27 * 135) = 40

 

135 - 45 - 30 - 20 = 40

 

40 = 40, the numbers check.

 

Questions?