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October 7th, 2018, 04:25 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, 05:21 PM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,664 Thanks: 2644 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. Last edited by v8archie; October 7th, 2018 at 05:50 PM. 
October 7th, 2018, 05:46 PM  #3 
Global Moderator Joined: Dec 2006 Posts: 20,758 Thanks: 2138  
October 7th, 2018, 07:32 PM  #4  
Member Joined: Sep 2018 From: Japan Posts: 35 Thanks: 2  Quote:
 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. v8archie, Thank you for this  
October 8th, 2018, 02:23 AM  #5 
Global Moderator Joined: Dec 2006 Posts: 20,758 Thanks: 2138  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, 04:47 AM  #6 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,664 Thanks: 2644 Math Focus: Mainly analysis and algebra 
There's clearly some language barrier here, but my Japanese is nonexistent. I think I now understand your idea, and it seems to be fine.

October 9th, 2018, 04:07 PM  #7  
Member Joined: Sep 2018 From: Japan Posts: 35 Thanks: 2  Quote:
Hehe  
October 9th, 2018, 04:13 PM  #8 
Global Moderator Joined: Dec 2006 Posts: 20,758 Thanks: 2138 
The correct multiplier is (11  7)/(7  5) = 4/2 = 2.

October 9th, 2018, 05:25 PM  #9 
Member Joined: Sep 2018 From: Japan Posts: 35 Thanks: 2  

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pencils, pens, ratio 
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