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

kingscrips

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

I just can't figure out how to solve it. Please help!

Here it goes...

" Steve, Mark, Jason and Emily shared the cos of a present equally. Mark did not bring enough money and only managed to pay for half his share. Only Steve and Jason helped to pay for the rest of Mark's share. The ratio of the amount of money Steve paid to the amount of money Jason paid was 3:4. The next day, Mark returned $24 to Jason. How much did the present cost?"

Thanks in advance...
 

MarkFL

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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.
 

kingscrips

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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

other $18.

That means Jason paid total of $108 and Steven paid $102.

But the question said they paid in ratio of 3:4. Can you explain if I am getting this right?

Thank you.
 

MarkFL

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Yes, each person's share was $84. Mark was short half of that, or $42. Steve and Jason covered this in a ratio of 3:4. That means Steve covered 3/7 and Jason covered 4/7. 4/7 of 42 is 24, so everything checks out. :)
 

kingscrips

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Yes, each person's share was $84. Mark was short half of that, or $42. Steve and Jason covered this in a ratio of 3:4. That means Steve covered 3/7 and Jason covered 4/7. 4/7 of 42 is 24, so everything checks out. :)
Mark,

I have one more question.. "The ratio of the amount of money Steve paid to the amount of money Jason paid was 3:4 "

My understanding to above sentence was total amount Steve and Jason paid towards present. If that's the case is there different answer to this

question?

Thank you so much!!
 

Otis

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… My understanding to above sentence was total amount Steve and Jason paid towards present. If that's the case is there different answer to this question?
No, there's only one answer that works at the Pre-Algebra level. (See MarkFL's next post, for a high-school algebra approach to a different interpretation of the exercise.)

Maybe you're thinking about this sentence: The ratio of the amount of money Steve paid to the amount of money Jason paid was 3:4.

If so, I wondered what it meant, too. I had to confirm that only one of the interpretations works. In other words, I had to start doing the exercise, in order to figure out the meaning of that sentence. The exercise could be better worded:

Steve and Jason each paid extra, to cover the rest of Mark's share. The ratio of the extra amount Steve paid to the extra amount Jason paid was 3:4.

😎

EDIT: Added comment comparing pre-algebra to high-school algebra.
 
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MarkFL

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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:

\(\displaystyle P\left(\frac{1}{4}+\left(1-\frac{1}{7}\right)\frac{1}{8}\right)=24\)

\(\displaystyle P\left(\frac{1}{4}+\frac{6}{7}\cdot\frac{1}{8}\right)=24\)

\(\displaystyle P\left(\frac{1}{4}+\frac{3}{28}\right)=24\)

\(\displaystyle P\left(\frac{5}{14}\right)=24\)

\(\displaystyle P=24\cdot\frac{14}{5}=67.2\)
 

kingscrips

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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:

\(\displaystyle P\left(\frac{1}{4}+\left(1-\frac{1}{7}\right)\frac{1}{8}\right)=24\)

\(\displaystyle P\left(\frac{1}{4}+\frac{6}{7}\cdot\frac{1}{8}\right)=24\)

\(\displaystyle P\left(\frac{1}{4}+\frac{3}{28}\right)=24\)

\(\displaystyle P\left(\frac{5}{14}\right)=24\)

\(\displaystyle P=24\cdot\frac{14}{5}=67.2\)
Thank you again with your detailed explanation. The answer I was told is $224. Maybe $224 is wrong?
 

MarkFL

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Who told you the answer is $224?
 

kingscrips

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Teacher told my son the answer is $224. How can we make an equation base on this answer?
 

MarkFL

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If the present cost $224, then each of the 4 people are responsible for $56. Mark would then need half of that covered or $28. 4/7 of that is $16, not $24 if we assume my original interpretation of the problem is correct.
 

kingscrips

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$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

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$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

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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

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Thank you Mark!!

Now I just got to explain it to my son... :LOL::LOL::LOL:
 

Otis

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… 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

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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

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… [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
 
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