My Math Forum Ratio of pens and pencils

 Elementary Math Fractions, Percentages, Word Problems, Equations, Inequations, Factorization, Expansion

 October 7th, 2018, 05:25 PM #1 Member   Joined: Sep 2018 From: Japan Posts: 35 Thanks: 2 Ratio of pens and pencils The ratio of the number of pens to the number of pencils in a box was 7:5. After 21 pens and 21 pencils were sold, the ratio became 11:7. How many pencils were left at the stall? My work: Ratio (before) Pens:Pencils 7:5 Ratio (after) Pens:Pencils 11:7 Since the ratio of both pens and pencils were sold, I twice the ratio for pens and pencils for before. Below is what I did Ratio (before) Pens:Pencils 7:5 * 2 => Pens:Pencils 14:10 Then I subtracted 14 - 11 (the 11 is the after ratio) and 10 - 7 (the 7 is the after ratio) and got 3 units Since we know the difference for pens and pens is 21, below is what I did. 3 units = 21 => 1unit = 7. How many pencils were left at the stall? 7 * 7 = 49 pencils were left. Is my work correct?
 October 7th, 2018, 06:21 PM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,598 Thanks: 2583 Math Focus: Mainly analysis and algebra \begin{align} \text{Pens at end} &= p \\ \text{Pencils at end} &= q \\[12pt] \text{after:} \; \frac p q &= \frac{11}7 \\ \text{before:} \; \frac{p+21}{q+21} &= \frac75 \end{align} That's two equations in two variables that you can solve by your favourite method. Your answer is correct. I don't recognise the method, but it seemed to work. Thanks from xoritos Last edited by v8archie; October 7th, 2018 at 06:50 PM.
October 7th, 2018, 06:46 PM   #3
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Quote:
 Originally Posted by xoritos Is my work correct?

October 7th, 2018, 08:32 PM   #4
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Quote:
 Originally Posted by skipjack Your method and working were correct, but your explanation of why you did your first step of your solution doesn't make sense.
is this the one you are talking about?
------------------------------------------------
Since the ratio of both pens and pencils were sold, I twice the ratio for pens and pencils for before. Below is what I did

Ratio (before)
Pens:Pencils
7:5 * 2 =>

Pens:Pencils
14:10

----------------------------------------------

This is the part where I wasn't sure if it's mathematically correct, or what I explained was mathematically correct. What I did was since both ratios of pens and pencils changed. Since there is not equivalent ratio, I multiplied both the pens and pencils ratio by 2 so I can subtract the after ratio (11:7) from the modified before ratio (14:10) to get my single unit.

From what I understand with these ratio problems and using units to get an answer. The unit that doesn't change is fixed.

Quote:
 Originally Posted by v8archie \begin{align} \text{Pens at end} &= p \\ \text{Pencils at end} &= q \\[12pt] \text{after:} \; \frac p q &= \frac{11}7 \\ \text{before:} \; \frac{p+21}{q+21} &= \frac75 \end{align}
v8archie, Thank you for this

October 8th, 2018, 03:23 AM   #5
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Quote:
 Originally Posted by xoritos . . . Since there is not equivalent ratio, I multiplied both the pens and pencils ratio by 2 so I can . . .
The method is valid and your working was correct, but your justification for multiplying by 2 was "there is not equivalent ratio", which doesn't clearly explain why you chose to multiply by 2 rather than by some other number.

 October 8th, 2018, 05:47 AM #6 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,598 Thanks: 2583 Math Focus: Mainly analysis and algebra There's clearly some language barrier here, but my Japanese is non-existent. I think I now understand your idea, and it seems to be fine. Thanks from xoritos
October 9th, 2018, 05:07 PM   #7
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Quote:
 Originally Posted by skipjack The method is valid and your working was correct, but your justification for multiplying by 2 was "there is not equivalent ratio", which doesn't clearly explain why you chose to multiply by 2 rather than by some other number.
You do have a good point. If I multiplied by 3, I would get a before ratio of (21:15). Then If I subtracted from the after ratio, I would get 21 - 11 = 10 units and 15 - 7 = 8 units. Which wouldn't work when dividing with 21. The way how I did it seems to work, but I'm not fully sure why.

Quote:
 Originally Posted by v8archie There's clearly some language barrier here, but my Japanese is non-existent. I think I now understand your idea, and it seems to be fine.
Hehe

 October 9th, 2018, 05:13 PM #8 Global Moderator   Joined: Dec 2006 Posts: 20,262 Thanks: 1958 The correct multiplier is (11 - 7)/(7 - 5) = 4/2 = 2. Thanks from v8archie and xoritos
October 9th, 2018, 06:25 PM   #9
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Quote:
 Originally Posted by skipjack The correct multiplier is (11 - 7)/(7 - 5) = 4/2 = 2.
Thank you for this. That made it more sense

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