# Word question about ratio: Steve, Mark, Jason and Emily shared the cost of a present equally....

#### kingscrips

##### New member
Hi, I am trying to solve word question for my 5th grade son.

Here it goes...

#### kingscrips

##### New member
Hello, and welcome to FMH!

Normally we don't provide complete solutions, but I tend to do so when it is a parent helping their child. I would encourage you to guide your son and use the help I provide as a series of hints to your son to get him to do the heavy lifting.

The amount that Mark was short was 1/8 of the cost of the present (1/2 of his 1/4 share). Jason paid 4/(3 + 4) = 4/7 of that, which would be (1/8)(4/7) = 1/14 of the cost of the present, and we are told this was $24. Thus, the cost of the present would be 14 times$24 or $336. Hello Mark, Thanks for your quick reply. I was going over your answer and I got little confused. Base on your answer each person paid$84. (84*4= $336) If Mark paid half at first he paid$42. Remaining $42 was split between Steve and Jason. If Mark gave$24 to Jason then Mark also needs to pay Steve

#### kingscrips

##### New member
$56 for each person. Mark need$28 covered. Jason paid $24 and Steve paid$4. 56+4=60 and 56+24=80. 3:4 ratio.

I understood this part. Now I am stuck on how to solve this and explain it to my son.

#### Otis

##### Senior Member
$56 for each person. Mark need$28 covered. Jason paid $24 and Steve paid$4. 56+4=60 and 56+24=80. 3:4 ratio.

I understood this part. Now I am stuck on how to solve this and explain it to my son.
Off the top of my head, I can't think of a fifth-grade arithmetic explanation, for the alternate interpretation above. I had to use algebra (solving a system of two equations), to get $224. Maybe I'll wake up with a different idea … #### MarkFL ##### Super Moderator Staff member I made an error in my post regarding the second interpretation...here's what I should have posted: I interpreted the 3:4 ratio as pertaining to Mark's shortfall they helped cover. If we interpret it as you suggest, then: The amount that Mark was short was 1/8 of the cost of the present (1/2 of his 1/4 share). Steve and Jason covered this shortfall. Let k be the portion Steve covered and so 1 - k would be the portion Jason covered. And so we could write: $$\displaystyle \frac{\frac{1}{4}+k\frac{1}{8}}{\frac{1}{4}+(1-k)\frac{1}{8}}=\frac{3}{4}$$ $$\displaystyle 1+\frac{1}{2}k=\frac{3}{4}+\frac{3}{8}(1-k)$$ $$\displaystyle 8+4k=6+3(1-k)$$ $$\displaystyle 8+4k=6+3-3k$$ $$\displaystyle 7k=1$$ $$\displaystyle k=\frac{1}{7}$$ Hence, letting $$P$$ be the cost of the present: $$\displaystyle P\left(\left(1-\frac{1}{7}\right)\frac{1}{8}\right)=24$$ $$\displaystyle P\left(\frac{6}{7}\cdot\frac{1}{8}\right)=24$$ $$\displaystyle P\left(\frac{3}{28}\right)=24$$ $$\displaystyle P=24\cdot\frac{28}{3}=224\quad\checkmark$$ Sorry for the confusion. #### kingscrips ##### New member Thank you Mark!! Now I just got to explain it to my son... #### Otis ##### Senior Member … Now I just got to explain it to my son Are you going to explain the algebra or attempt to adapt the algebra to an arithmetic approach? You posted on the Pre-Algebra board. If your son is not studying algebra, then I'd be interested to see the arithmetic explanation of this fifth-grade exercise. Please consider posting that, once you're finished. Cheers! #### kingscrips ##### New member Ok We tried our best to explain this to our son and he understood.. Steve Mark Jason Emily X+(1/2X - 24) 1/2X X + 24 X 3 : 4 = Steve : Jason 3 : 4 = X+(1/2X - 24) : X + 24 3( X + 24) = 4 { X + (1/2X - 24) } 3X + 72 = 4X + ( 2X - 96) 3X + 72 = 6X - 96 72 + 96 = 6X - 3X 168 = 3X 56 = X Each person paid$56

Present cost 56 x 4 = \$224

#### Otis

##### Senior Member
… [explained the algebra] to our son and he understood …
Sounds like your son is way ahead of the curve. Good for you!

Next time he needs help, he could post his work and then we could work with him directly.

Cheers