Fraction 'word' question

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nav1

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Hi All,

My first post, so apologies if any errors made.

I have a question about fractions. It's age 12-13 maths level, I'm from the UK and this is just me doing maths for personal development

It's Q2 I'm stuck on. I managed Q1 fine.

The way Q2 is worded I don't understand. Any help would be greatly appreciated.
IMG-2372.jpg
Thank you!
 
It's asking what [MATH]\frac{3}{7}[/MATH] of [MATH]\frac{7}{15}[/MATH] of [MATH]30[/MATH] is. Using the impractical, convoluted, "trying a bit too hard" style of a typical word problem.

It also doesn't state how many of the non-vegetarian sandwiches have cheese. Or how cheese is a vegetable. Maybe they should have just stuck to the numbers?
 
Mr. Bland said:
It's asking what [MATH]\frac{3}{7}[/MATH] of [MATH]\frac{7}{15}[/MATH] of [MATH]30[/MATH] is. Using the impractical, convoluted, "trying a bit too hard" style of a typical word problem.

It also doesn't state how many of the non-vegetarian sandwiches have cheese. Or how cheese is a vegetable. Maybe they should have just stuck to the numbers?
Click to expand...

Hi Mr. Bland,

Thanks for you response.

I managed to figure out the answer (6).

I've attached a photo of my workings' for your (and others) perusal.
Q2 Workings.jpg
Any comments/recommendations are greatly appreciated.

Thank you!
 
You got it! The problem focuses on fractions with different denominators, and still it works out well even though the denominators don't all share common factors. Despite [MATH]\frac{1}{7}[/MATH] and [MATH]\frac{1}{15}[/MATH] not having a lot in common when it comes to conversions, that didn't actually complicate things when working through the problem one step at a time.

If you're feeling energetic (or maybe lazy?), there's a convenient rule of thumb when it comes to the word "of": it almost always describes multiplication. Since the original problem was finding "[MATH]\frac{3}{7}[/MATH] of [MATH]\frac{7}{15}[/MATH] of [MATH]30[/MATH]", you can literally write it like this:

[MATH]\frac{3}{7} * \frac{7}{15} * 30 = \frac{3 * 7 * 30}{7 * 15 * 1} = \frac{630}{105} = 6[/MATH]​

In the same way, the word "per" generally describes division (kilometers per hour, etc.). These mental shortcuts can greatly reduce the effort needed to solve a word problem.
 
nav1 said:
Hi Mr. Bland,

Thanks for you response.

I managed to figure out the answer (6).

I've attached a photo of my workings' for your (and others) perusal.
View attachment 22622
Any comments/recommendations are greatly appreciated.

Thank you!
Click to expand...
I do have some problems with your work although you reached the correct answer. 1st of please do not us an arrow for an equal sign. It is 7/15 = 14/30.

You say that 14/30 sandwiches are veg. You do realize that 14/30 is less than 1, correct? But you know that 14 sandwiches are veg so why write 14/30 are veg? Now maybe you meant 14 out of 30 sandwiches are veg..

You know that 3/7 of the veg sandwiches contain cheese. That is you know 3/7 of the 14 veg sandwiches contain cheese. Then (3/7)*14 = 6 sandwiches which contain cheese.

Sometimes, maybe in a different type problem, there will not be a whole number of cheese sandwiches. That is it will not be easy to write 3/7 with a 14 in the denominator. For example what if 3/7 of the 15 out of 30 veg sandwiches had cheese? You can not easily write 3/7 with a 15 in the denominator. Why not just compute (3/7)*15 = 45/7 = 6 3/7.
 
Mr. Bland said:
You got it! The problem focuses on fractions with different denominators, and still it works out well even though the denominators don't all share common factors. Despite [MATH]\frac{1}{7}[/MATH] and [MATH]\frac{1}{15}[/MATH] not having a lot in common when it comes to conversions, that didn't actually complicate things when working through the problem one step at a time.

If you're feeling energetic (or maybe lazy?), there's a convenient rule of thumb when it comes to the word "of": it almost always describes multiplication. Since the original problem was finding "[MATH]\frac{3}{7}[/MATH] of [MATH]\frac{7}{15}[/MATH] of [MATH]30[/MATH]", you can literally write it like this:

[MATH]\frac{3}{7} * \frac{7}{15} * 30 = \frac{3 * 7 * 30}{7 * 15 * 1} = \frac{630}{105} = 6[/MATH]​

In the same way, the word "per" generally describes division (kilometers per hour, etc.). These mental shortcuts can greatly reduce the effort needed to solve a word problem.
Click to expand...
Thank you Mr Bland. Excellent explanation and insight! Much appreciated.
 
T
Jomo said:
I do have some problems with your work although you reached the correct answer. 1st of please do not us an arrow for an equal sign. It is 7/15 = 14/30.

You say that 14/30 sandwiches are veg. You do realize that 14/30 is less than 1, correct? But you know that 14 sandwiches are veg so why write 14/30 are veg? Now maybe you meant 14 out of 30 sandwiches are veg..

You know that 3/7 of the veg sandwiches contain cheese. That is you know 3/7 of the 14 veg sandwiches contain cheese. Then (3/7)*14 = 6 sandwiches which contain cheese.

Sometimes, maybe in a different type problem, there will not be a whole number of cheese sandwiches. That is it will not be easy to write 3/7 with a 14 in the denominator. For example what if 3/7 of the 15 out of 30 veg sandwiches had cheese? You can not easily write 3/7 with a 15 in the denominator. Why not just compute (3/7)*15 = 45/7 = 6 3/7.
Click to expand...
Thank you Jomo! Another excellent answer and further correction to my workings’. I’ll keep that in mind regarding if there isn’t a whole number of sandwiches to begin with.

Much appreciated,
Nav1
 
nav1 said:
T

Thank you Jomo! Another excellent answer and further correction to my workings’. I’ll keep that in mind regarding if there isn’t a whole number of sandwiches to begin with.

Much appreciated,
Nav1
Click to expand...
Even if there is not a whole number of sandwiches your work is not exact. Again, there are NOT 14/30 veg sandwiches. This simply is not true. There are 14 veg sandwiches. All I said in my previous post is that your incorrect method will not get you the correct solution if there will be a fraction of sandwiches in the problem. Either way your method was not correct! Mr Bland was being generous saying that you were correct.
 
After the first step, [MATH]\frac{14}{30}[/MATH] of the sandwiches were vegetarian. The exact literal wording of "[MATH]\frac{14}{30}[/MATH] sandwiches are vegetarian" is technically incorrect, but the intended meaning is clear from the context. The part that matters is understanding how the proportions relate to one another, and this understanding has been demonstrated.

That being said, do be mindful of how you write things down. If there's potential for ambiguity, your instructors or TAs may not be very forgiving. Mathematics is a field of study where pedantry can genuinely be a matter of life-or-death, so mathematicians tend to prefer the "exact literal technically correct" side of things.
 
Thanks for your responses. I will study the recommendations hard and try my best to improve my understanding of the problems.
 
nav1 said:
Thanks for your responses. I will study the recommendations hard and try my best to improve my understanding of the problems.
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You did provide the correct answer - "...6 sandwiches are cheese". We split hair about the actual statement (hair in sandwich?!!) - but the answer was very clear.

So you might get a grade of A- instead of A+. And sometimes - good enough is good enough.
 
I did like your thought process. It was quite clear, actually impressive. I think that somone who can think thru a problem like that can also write things clearer than you did. That was why I was so hard on you. Please take my being hard on you as a compliment.
 
HallsofIvy said:
A lot of vegetarians would NOT consider a "cheese" sandwich to be vegetarian!
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Those would be "Vegan" - extreme vegetarians - no milk (product).

And there are Eggitarians - who eat eggs but no meat products.
 
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