# Thread: Need help with a fractions problem

1. ## Need help with a fractions problem

For the question

"Juanita made two sandwiches. She cut the sandwiches into 4 pieces each. She has 5 pieces left. What fraction of the sandwiches are left?"

Shouldn't the answer be 5/8? Yet in my math textbook it says the answer is 1 1⁄4. I've attached a scan of the textbook page.

It seems that both answers make sense, yet if that were true, then 5/8 would equal to 1 1⁄4 but that is obviously not the case, so I'm really confused. Could someone explain to me which answer is correct? Thanks in advance!

IMG_20171205_0002.jpg

2. Interesting. I think I could go either way, but the question is poorly worded.

1) What portion of "the sandwiches" remains? 5/8

2) What portion or EACH sandwhich remains? 1, 1/4

3) How many sandwiches remain? 1 + 1/4 = 5/4

Seriously, I'd complain about this one. Very unclear. It appears to want #3, but it seems to me that the words are asking for #1.

3. Originally Posted by palespring
For the question

"Juanita made two sandwiches. She cut the sandwiches into 4 pieces each. She has 5 pieces left. What fraction of the sandwiches are left?"

Shouldn't the answer be 5/8? Yet in my math textbook it says the answer is 1 1⁄4. I've attached a scan of the textbook page.

It seems that both answers make sense, yet if that were true, then 5/8 would equal to 1 1⁄4 but that is obviously not the case, so I'm really confused. Could someone explain to me which answer is correct? Thanks in advance!

IMG_20171205_0002.jpg
There are five quarters left. How much is five quarters of a sandwich? Clearly, 1 and 1 quarter of a sandwich.

I agree with tkhunny. The question is very badly worded. What is intended is to ask what fraction of a sandwich is left. Five quarter sandwiches do represent five eighths of eight original quarter sandwiches, but those eight quarters represented TWO sandwiches, not one.

$\dfrac{5}{8} * 2 = \dfrac{5}{4} = 1 + \dfrac{1}{4}.$

But the use of the plural "sandwiches" completely destroys the intended meaning.

4. Originally Posted by palespring
"Juanita made two sandwiches. She cut the sandwiches into 4 pieces each. She has 5 pieces left. What fraction of the sandwiches are left?"
I fully agree with tkhunny and JeffM. "The sandwiches" should mean "the total amount she started with", and 5/8 of that amount is what is left. There is also a grammatical error: the subject of this sentence is "what fraction", so the verb should be singular, "is left", not "are left".

One way to explain the issue is that when you talk about a fraction in any applied problem, you have to identify the "unit": what is being thought of as "the whole" or "1". This is indicated in English by "of": a fraction of the whole. In the question (both the form you quote and the later version, "What fraction of the sandwiches does Juanita have left?"), the unit seems to be "the sandwiches". The proper phrasing for the question they intended to ask (in which the unit is one sandwich) is, "What fraction of a sandwich", or just "How many sandwiches".

I think the authors were so focused on the answer (introducing the idea of a mixed number) that they forgot to write the question carefully. (Actually, it seems that they used the plural verb because they knew what their answer was going to be: more than one. But "the fraction of the sandwiches" that is left has to be less than one.)

I hope they make things clearer going forward, especially in problems you have to solve yourself. But good for you in recognizing the correct answer.

5. Thanks for the reply everyone! I was also thinking that the question was wrongly worded but wasn't sure, and now I am.

Originally Posted by Dr.Peterson
Actually, the only person who can really clarify doubts about the meaning of a question is the author of the question! If you are unsure of the meaning of a question posed by your teacher, ask your teacher. Don't ask some third party.
I'm actually an adult self-teaching out of a textbook so I don't have a teacher. Thanks for your input!

6. Originally Posted by palespring
I'm actually an adult self-teaching out of a textbook so I don't have a teacher. Thanks for your input!
I was wondering about your context -- you could have been a child, a parent, a teacher, or a future teacher. Sometimes it can be helpful to know that.

I hope we haven't beaten the subject to death. It is worth knowing that textbooks aren't perfect, so sometimes when you disagree, you may be right. Especially as an adult, you can think of authors as fellow-humans, but still learn from them. And as I said, others like us, though "third parties", can be helpful in recognizing errors so you can move on, even though we can only guess what the author was thinking. (Or maybe it was an editor.)

7. Originally Posted by Dr.Peterson
It is worth knowing that textbooks aren't perfect, so sometimes when you disagree, you may be right. Especially as an adult, you can think of authors as fellow-humans, but still learn from them. And as I said, others like us, though "third parties", can be helpful in recognizing errors so you can move on, even though we can only guess what the author was thinking. (Or maybe it was an editor.)
Yep, agreed. I've encountered a few errors in this textbook before so I've learned not to have blind faith in the book's accuracy. And this forum has been very helpful in clarifying and validating my concerns so again I appreciate everyone's input.

8. ## Mixed number and Improper fractions

Improper fractions are when numerator is greater than denominator.........eg:5/4,9/7

Mixed number is a number consisting of a whole number and a proper fraction. eg: 1 1/4, 2 3/4

In your case Juanita has cut each sandwich to 4, if 5 pieces are left, then the mixed number is 1 and 1/4 sandwich is left or 5/4 sandwich is left ............it is not 5/8 left.

If Juanita cut each sandwich to 8 pieces each, and 10 pieces are left, then the mixed number or sandwich left is 1 and 2/8 or as improper fraction 10/8 .
If Juanita had cut each sandwich to 5 pieces each, and 8 pieces are left, then the sandwich left in terms of mixed number is 1 and 3/8 and as improper fraction it is 8/5.
If Juanita had cut each sandwich to 3 pieces each and 5 pieces are left, then the sandwich left in terms of mixed numbers is 1 and 2 /3 and improper fraction is 5/3.

Eg below:This is 1 and 3/4 as a mixed number. It can be written as 7/4 as a improper fraction so 1 and 3/4 pizzas are left or 7/4 pizzas are left....................its not 7/8.

9. Originally Posted by palespring
"Juanita made two sandwiches. She cut the sandwiches into 4 pieces each. She has 5 pieces left. What fraction of the sandwiches are left?"

Shouldn't the answer be 5/8?

Why would it? One "whole" is one sandwich. She cut each sandwich into four pieces, giving her 8/4 (pieces of) sandwiches (which simplies to "2", as sense would require). If she now has five pieces, then these are five of the one-fourth-of-a-sandwich pieces, which is 5/4 of a sandwich (or one and one-fourth).

10. Originally Posted by stapel
Why would it? One "whole" is one sandwich. She cut each sandwich into four pieces, giving her 8/4 (pieces of) sandwiches (which simplies to "2", as sense would require). If she now has five pieces, then these are five of the one-fourth-of-a-sandwich pieces, which is 5/4 of a sandwich (or one and one-fourth).[/SIZE]
@stapel

The problem is that the question did not ask "How much of a sandwich is left." It asked "How much of the sandwiches are left."

What do we find in the first sentence: an indefinite article referring to a standard or perhaps platonically ideal sandwich, a singular noun consistent with the indefinite article, and a singular verb consistent with a singular subject. What do we find in the second sentence, which is the actual question that the student is asked to parse: a definite article (referring, according to the rules of English grammar, to previously identified exemplars of the indicated class), a plural noun (consistent with two sandwiches having been cut), and a plural verb consistent with a plural subject.

Others have been kinder than I am and have said that the wording of the problem is ambiguous. I disagree. According to the basic rules of English grammar, "the sandwiches are" indicates specific sandwiches identifiable by context, not an abstract idealized sandwich. The only correct answer to the question asked is 5/8 of the two original sandwiches. The answer given by the text book is to a question that the text failed to ask.

It is all very well to say that is a stupid question. I agree. Probably, the author of the text book meant "How much of a sandwich is left." But it is ridiculous to blame the reader's confusion on the author's inability to express that intended meaning in English. The value of word problems is that they require translating issues presented in a natural language into the language of mathematics, a skill critical to applying mathematics to any practical purpose. The least we should require (although obviously not anything we can expect from the current educational establishment) is that the word problems posed to English-speaking students of arithmetic follow the rules of English grammar, not some language that looks like English but follows some grammar known only to the author.

One of the problems in following this thread is that, for perfectly good reasons, a chunk of it has been transferred into a separate thread.

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