Homework help - hard fraction question

[einstein]

Hard fraction questions:

a) What must 2 1/7 be divided by to give 1 3/7 ?
b) What must 3 4/5 be multiplied by to give 10 2/5?

It's a pre-algebra, no calculator allowed question.

Do i have to just keep guessing until i get the answers? Or are there exact formulas for solving these problems? (You cant use an algebra formula because its from a pre-algebra book and algebra has not even been covered yet)

 

[Denis]

a) What must 2 1/7 be divided by to give 1 3/7 ?

2 1/7 = 15/7
1 3/7 = 10/7

(15/7) / x = 10/7
x = (15/7) / (10/7)
x = (15/7) * (7/10)

Finish it

 

[einstein]

a) What must 2 1/7 be divided by to give 1 3/7 ?

2 1/7 = 15/7
1 3/7 = 10/7

(15/7) / x = 10/7
x = (15/7) / (10/7)
x = (15/7) * (7/10)

Finish it

Thanks but like i mentioned above, you cannot use algebra to solve this question. This is from a year 6 (12 year olds) pre algebra book - they do not know algebra yet so you cannot expect them to magically know algebra to solve this problem. I already know how to solve it using algebra. But i want to know how the book expects you to solve it without using algebra?

 

[Mrspi]

Hard fraction questions:

a) What must 2 1/7 be divided by to give 1 3/7 ?
b) What must 3 4/5 be multiplied by to give 10 2/5?

It's a pre-algebra, no calculator allowed question.

Do i have to just keep guessing until i get the answers? Or are there exact formulas for solving these problems? (You cant use an algebra formula because its from a pre-algebra book and algebra has not even been covered yet)

Ok... no algebra. But, I sure hope "common sense" is all right!

2 1/7 divided by SOMETHING gives 1 3/7

Now, from basic arithmetic (which comes BEFORE pre-algebra) you're expected to know that the answer to a division problem, multiplied by the divisor, should give you the number you were dividing into.

Example....
6 divided by 2 gives 3
Check: divisor * quotient = dividend
(number you were dividing by * answer = number you were dividing into)
2 * 3 = 6
6 = 6
Checks!!

Ok....
2 1/7 divided by SOMETHING gives 1 3/7
Let's write both 2 1/7 and 1 3/7 as improper fractions (this is covered before pre-algebra, too):
15/7 divided by SOMETHING gives 10/7

And the checking process tells us that

Answer * divisor = number we're dividing into
10/7 * ???? = 15/7

When you multiply fractions, you multiply the numerators together, and you multiply the denominators together.

So 10 * (something) = 15
and 7 * (something else) = 7

Apparently the "something else" must be 1, because 7*1 is 7

And 10*(something) = 15

Now I'm tired....a bit of algebra would make this SO MUCH easier.

But you asked for an explanation without algebra, and I've tried.

10 * something = 15
I guess the "something" must be greater than 1....

Good luck.

 

[einstein]

Thanks for that it made sense to me.

The only problem is being able to come up with that on my own. I think i need to go back and study the properties of division and multiplication.

 

[BigGlenntheHeavy]

a) What must 2 1/7 be divided by to give 1 3/7 ?

\(\displaystyle Now \ 2\frac{1}{7} \ and \ 1\frac{3}{7} \ are \ mixed \ numbers.\)

\(\displaystyle Hence, \ changing \ them \ to \ improper \ fractions, \ we \ get: \ \frac{15}{7} \ and \ \frac{10}{7}\)

\(\displaystyle So \ now \ our \ question \ becomes, \ "What \ must \ \frac{15}{7} \ be \ divided \ by \ to \ give \ \frac{10}{7}?"\)

\(\displaystyle \frac{15}{7} \ divided \ by \ ? \ equals \ \frac{10}{7}?\)

\(\displaystyle Or \ \frac{\frac{15}{7}}{?} \ = \ \frac{10}{7}.\)

\(\displaystyle Cross \ multiplying \ gives \ (10)(?) \ = \ \bigg(\frac{15}{7}\bigg)(7), \ \implies \ (10)(?) \ = \ 15 \ \implies \ ? \ = \ \frac{15}{10} \ = \ \frac{3}{2}.\)

\(\displaystyle Hence \ if \ we \ divide \ \frac{15}{7} \ by \ \frac{3}{2}, \ then \ we \ will \ get \ \frac{10}{7}.\)

 

[mmm4444bot]

I think i need to go back and study the properties of division and multiplication.

You might find THIS SITE useful for learning how to work with mixed numbers and improper fractions. There is also a link (near the bottom left) for other topics in pre-algebra.

 

[einstein]

Re:

I think i need to go back and study the properties of division and multiplication.

You might find THIS SITE useful for learning how to work with mixed numbers and improper fractions. There is also a link (near the bottom left) for other topics in pre-algebra.

Thanks for the link. I have already read that entire site! I have a very good understanding of fractions already.

Anyway, i still don't totally "get" how to do this problem without using algebra. Some concepts still obviously aren't clear to me. I think the way everyone is solving it above it by knowing that multiplying is the inverse of dividing, and visa-versa. And thats how you get the solution. But if you put this question in a test i don't think i'd figure it out myself. Its so frustrating. Im trying to figure out the gap in my knowledge that is hindering me.

 

[einstein]

\(\displaystyle Or \ \frac{\frac{15}{7}}{?} \ = \ \frac{10}{7}.\)

I get lost at this part. How do you arrive at that. What do i need to study to see what you have done here?

 

[Denis]

\(\displaystyle Or \ \frac{\frac{15}{7}}{?} \ = \ \frac{10}{7}.\)

I get lost at this part. How do you arrive at that. What do i need to study to see what you have done here?

That's simply rewriting the original problem: What must 2 1/7 be divided by to give 1 3/7 ?

It's clear that if a/b = c, then a/c = b, right?

So (15/7) / (10/7) = 3/2 is answer, and really has NO algebra involved.

 

[mmm4444bot]

i still don't totally "get" how to do this problem

Answer the following question.

What must 10 be divided by to give 5 ?

Next, think about HOW you determined the answer, and then explain your step(s) to me.

 

[einstein]

It's clear that if a/b = c, then a/c = b, right?

Thanks! I think that is what IS NOT clear in my head. How do you arrive at this fact a/b = c, then a/c = b? This is exactly the concept i need to get clear in my head, i think.

It isn't clear to me that if a/b = c, then a/c = b. How does this work? Is there any lesson or book that covers this in detail?

E.g. what is your understanding of a/b = c, then a/c = b, how do you arrive at it. Is it just something you've memorized, or do you know something deeper about it that i need to know? Why does it work?

I had a think about it last night until my brain was sore...

2 x 3 = 6
3 x 2 = 6

Because 2 and 3 are both factors of six, when you divide 6 by them, which ever one is the divisor the other will be the quotient. Is that correct?

Therefore:
6 / 2 = 3
6 / 3 = 2

And therefore a/b = c, and a/c = b. Is that right?

Am i correct? How can i go deeper on this or get this concept crystal clear in my head?

 

[einstein]

Re:

i still don't totally "get" how to do this problem

Answer the following question.

What must 10 be divided by to give 5 ?

Next, think about HOW you determined the answer, and then explain your step(s) to me.

2.

I got that answer because i know that half of 10 is 5. And to arrive at half a number you divide by 2. So i took a shortcut. I dont think i used the method you wanted me to. The method, that if i knew, i would be able to solve my original question.

I think what you wanted me to do is divide 10 by 5, to arrive at the answer, but this seems confusing to me. I dont understand how it works. Some basic concept of division isnt making sense to me i think.

What steps do YOU take to determine the answer to your question above? What goes on inside your brain to solve it?

 

[mmm4444bot]

2.

I got that answer because i know that half of 10 is 5.

Exactly, and you know this because you've memorized the multiplication table. You know that 10 is two 5s.

2 × 5 = 10

If I had asked, instead, "What must 10 be divided by to give 2 ?", you would have said 5 because you know that 10 is five 2s.

5 × 2 = 10

Earlier, you wrote that you have a very good understanding of fractions. The following must make sense to you, since you've memorized the multiplication table.

10/2 = 5 and 10/5 = 2

This compound statement shows a pattern, and the pattern is always true when a numerator common to both parts is the product of the remaining two numbers (divisor and quotient).

27/3 = 9 and 27/9 = 3 because 3 × 9 and 9 × 3 both equal 27

44/11 = 4 and 44/4 = 11 because 11 × 4 and 4 × 11 both equal 44

5.0/2 = 2.5 and 5.0/2.5 = 2 because 2 and 2.5 are factors of 5.0

(1/4)/2 = 1/8 and (1/4)/(1/8) = 2 because 2 ad 1/8 are factors of 1/4

et cetera

The point is that knowing one part allows you to consider the other, and vice versa.

Let's go back to my previous question (in red):

What must 10 be divided by to give 5 ?

This question amounts to:

10/? = 5

You now know to think of the question this way:

10/5 = ?

That's the principle.

Whenever you get into a situation involving a relationship that amounts to:

Numerator/Divisor = Quotient

you can immediately view Numerator as a product of the factors Divisor and Quotient.

Divisor × Quotient = Numerator

and then swap the factors in the relationship.

What must 2 1/7 be divided by to give 1 3/7 ?

This question amounts to:

(2 1/7)/? = 1 3/7

Which you now know is:

(2 1/7)/(1 3/7) = ?

The numerator common to both of these equations is the product of the divisor and quotient.

(1 3/7) × ( ? ) = 2 1/7

If you want to go deeper than Numerator/Divisor = Quotient and Divisor × Quotient = Numerator, I can't help you.

 

[einstein]

Re:

2 × 5 = 10

If I had asked, instead, "What must 10 be divided by to give 2 ?", you would have said 5 because you know that 10 is five 2s.

5 × 2 = 10

Earlier, you wrote that you have a very good understanding of fractions. The following must make sense to you, since you've memorized the multiplication table.

10/2 = 5 and 10/5 = 2

This compound statement shows a pattern, and the pattern is always true when a numerator common to both parts is the product of the remaining two numbers (divisor and quotient).

27/3 = 9 and 27/9 = 3 because 3 × 9 and 9 × 3 both equal 27

44/11 = 4 and 44/4 = 11 because 11 × 4 and 4 × 11 both equal 44

5.0/2 = 2.5 and 5.0/2.5 = 2 because 2 and 2.5 are factors of 5.0

(1/4)/2 = 1/8 and (1/4)/(1/8) = 2 because 2 ad 1/8 are factors of 1/4

et cetera

The point is that knowing one part allows you to consider the other, and vice versa.

Let's go back to my previous question (in red):

What must 10 be divided by to give 5 ?

This question amounts to:

10/? = 5

You now know to think of the question this way:

10/5 = ?

That's the principle.

Whenever you get into a situation involving a relationship that amounts to:

Numerator/Divisor = Quotient

you can immediately view Numerator as a product of the factors Divisor and Quotient.

Divisor × Quotient = Numerator

and then swap the factors in the relationship.

What must 2 1/7 be divided by to give 1 3/7 ?

This question amounts to:

(2 1/7)/? = 1 3/7

Which you now know is:

(2 1/7)/(1 3/7) = ?

The numerator common to both of these equations is the product of the divisor and quotient.

(1 3/7) × ( ? ) = 2 1/7

If you want to go deeper than Numerator/Divisor = Quotient and Divisor × Quotient = Numerator, I can't help you.

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Thank you soooo much for breaking it down like that! That's just what i needed. I get it now! I can't thank you enough for that help. You explained it just the way i needed it to be explained.

(1/4)/2 = 1/8 and (1/4)/(1/8) = 2 because 2 ad 1/8 are factors of 1/4

2 and 1/8 are factors of 1/4. I get it, that they are. But is 2 a true factor of 1/4? The definition of factor i had in my head was a number that fits evenly into another (whole) number. But i guess that is wrong as is the case here, i can certainly see that 2 and 1/8 are factors of 1/4.

It must be because the definition of factors i had in my head was still at and elementary text book explanation level.

Like, i can see that, 1/2 and 2 are factors of 1. Even though 2 is great then 1.

 

[mmm4444bot]

The noun "factor" has more than one definition.

I'm using the definition that says anytime numbers are multiplied together, those numbers are called factors (no matter what they look like), and the result of the multiplication is a number called "their product".

Cheers

 

[einstein]

Re:

The noun "factor" has more than one definition.

I'm using the definition that says anytime numbers are multiplied together, those numbers are called factors (no matter what they look like), and the result of the multiplication is a number called "their product".

Cheers

Okay cool. Would you say that the basic concepts we just discussed (that you explained) of factors and a/b = c, a/c = b, c * b = a, b * c = a, are something you need to be familiar with and helpful for knowing and solving lots of math problems later on in math? Such as in algebra etc?

It seems to me they are, and are an important concept to grasp. Thanks!