Math is Hard: Orange you glad you’re not in fifth grade?

Here’s the answer to a problem that I admit that I cheated on. I got thrown by the idea that giving three oranges away means subtract – that’s where visualizing the transaction helps.

Problem
Shomika was helping her family pick oranges in their grove. She took some oranges home to share with her three friends. She gave 3 more than half to Jennifer. Angela got half of the remainder and 3 more. She gave Josie half of the remainder plus 3. When she got home, she only had 10 oranges left. How many did she have when she left the grove?

Solution
Start from the bottom and work your way up. For example, to be left with 10, she had to have had 26 when she met Josie, because she gave half away (26-13=13) then three more (13-3=10). And 26 is half of 58, etc. It’s the same problem with three iterations. So once you figure out the first problem – how many did she have before she met Josie – you apply the same logic to the rest.

The answer, if you don’t want to try it yourself:

In order for Shomika to be left with 10, she had to have had 26 when she met Josie, because she gave half of the 26 to Josie, which left 13, then she gave Josie 3 more, leaving her with 10.

26/2 - 3 =
  13 - 3 =
      10

In order for Shomika to be left with 26, she had to have had 58 when she met Angela, because she gave half of the 58 to Angela, which left 29, then she gave Angela 3 more, leaving her with 26.

58/2 - 3 =
  29 - 3 =
      26

In order for Shomika to be left with 58, she had to have had 122 when she met Angela, because she gave half of the 122 to Angela, which left 61, then she gave Angela 3 more, leaving her with 58.

122/2 - 3 =
   61 - 3 =
       58

Therefore, Shomika had 122 oranges when she left the grove. We can also infer that Jennifer is her BFF, and Shomika has strong arms. If an average orange weighs 6oz., then 122 oranges would weigh 45.75lbs (6 x 122)/16 = 45.75.

I started going down the algebraic route but stopped, because this was a question for a fifth-grader, who hadn’t had algebra yet. But this is the solution I would have gone for.

Another Way to Look At It

Since you’re going backwards, you just reverse the operations. Shomika had 10 after subtracting 3 and halving the prior amount. So to get to the prior amount, you add three then double that amount.

Or another way to look at it – what if Shomika wanted her oranges back? Her “friends” had posted a mean message on Facebook that she wasn’t supposed to see – but she did! Because Shomika is good friends with Tangiers, who is friends with Angela on Facebook, and he can see all her posts! And both Angela and Josie “liked” it!

So Shomika went to Josie’s house and took the extra 3 from her (now Shomika has 13), then doubled that amount, and now she has 26. Then she went to Angela and asked for the 3 back (now she has 29), then doubled that, and she has 58. Josie wasn’t home when Shomika got to her house, so she sneaked in to her kitchen through the back door (because she knows the back door is always open so the dog can go out and poop). She grabbed the flowery pink bag (that’s just like Josie!) of oranges next to the kitchen sink and ran. It was hard going, because she was already carrying around 22lbs of oranges in a paper bag. When she got home, she pulled three oranges out of the flowery bag (now she has 61), and counted double that to make sure she got all 122 oranges back. She didn’t though. Josie ate one. Josie is fat and picks her nose!

Or algebraically:

1/2x - 3 = 10
   x - 6 = 20
       x = 26

Then

1/2x - 3 = 26
   x - 6 = 52
       x = 58

And

1/2x - 3 =  58
   x - 6 = 116
       x = 122

A Simpler Version

The question is a lot simpler if you don’t include the plus-three. This much easier example will give you an idea for how to solve the problem:

Bellybop helped her family pick oranges in their grove, and afterwards, she took some oranges home to share with her friends. She gave half of the total oranges to Pollysnack. Silipsity got half of what was left of that. And she gave Hesponda half of what was left after that. When she got home, she only had 10 oranges left. How many did she have when she left the grove?

Working backwards from 10:

She had 20 when she met with Hesponda, who took 10.    (20 – 10 leaves 10).
She had 40 when she met with Silipsity, who took 20.       (40 – 20 leaves 20.
She had 80 when she met with Pollysnack, who took 40.  (80 – 40 leaves 40).

It’s a bit harder when you add the plus-three variant.

A Funner Version

Annabelle works for Apple Computer, and she was blackmailed into stealing iPads from the warehouse by three mafioso. The first guy, Guy (pronounced “Gee”), got half of the loot, plus seven extra to sell on eBay under the handle: “FlyGuy”. The next fella, Rosie, got half of what was left, plus 1. He wanted to post on Youtube a video of someone diving into a pool with a brand new iPad with the extra one (yeah, that was him!). Donny, the last guy, got half of the remainder, plus an extra 2 for his kids. Annabelle was left with 1, so if they got busted, she’d be holding the hot merch, too. How many iPads were stolen?

Work it backwards. The first calculation is for Donny:

1/2x - 2 = 1
   x - 4 = 2
       x = 6

Then Rosie:

1/2x - 1 = 6
   x - 2 = 12
       x = 14

And finally, Guy:

1/2x - 7 = 14
  x - 14 = 28
       x = 42

Next: English lesson on “had to have had”. Shomika had to have had had a lot of oranges. Also, why is it “backwards” as an adverb and “backward” as an adjective?

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