Comparing 2 Fraction based variables

In summary: Hi,I have a question with my original solution:i.e1/c = 1 + 1/dWhy can't we write:c= 1 + d?because it is incorrect. See below.which is obtained by taking reciprocal of both sides.The reciprocal of ##1 + \frac 1 d## is NOT 1 + d.Here is an example using numbers.$$\frac 1 {\frac 1 3} = 1 + \frac 1 {\frac 1 2}$$Does it follow that ##\frac 1 3 = 1 + \frac 1 2##?
  • #1
zak100
462
11

Homework Statement


c & d are positive integers &
1/c = 1 + 1/d
Now tell which is greater c or d?

Homework Equations


1/c = 1 + 1/d

The Attempt at a Solution


1/c = (d+1)/d

c = d/ (d+1)
c = 1 + d

Thus if c =10
therefore d = 9
But this is not a correct answer. Some body please guide me.

Zulfi.
 
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  • #2
Hint : Which is smaller 1/c or 1/d ?
 
  • #3
You could also just sub in values and make inferences from that
 
  • #4
For c, d positive integers, 1/c is always less than 1 unless c=1. I think the statement of the problem is incorrect that c and d are positive integers. There is no solution. c=1 implies d=+infinity. If the problem statement had said positive reals, then in one case c<1 and d>1. In another case, c<1 and d<1 and c<d, so that c<d if c, d are positive reals.
 
Last edited:
  • #5
zak100 said:

Homework Statement


c & d are positive integers &
1/c = 1 + 1/d
Now tell which is greater c or d?

Homework Equations


1/c = 1 + 1/d

The Attempt at a Solution


1/c = (d+1)/d

c = d/ (d+1)
c = 1 + d

Thus if c =10
therefore d = 9
But this is not a correct answer. Some body please guide me.

Zulfi.

As pointed out in #4, the equation is impossible if c and d are positive integers; however, if they are just positive real numbers, then the equation says that 1/c is a bigger number than 1/d. What does that say about c and d themselves?
 
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  • #6
Hi,
Thanks for your help. Actually it says that " c & d are positive".

<Which is smaller 1/c or 1/d ?>
I think it depends on values. They can be equal.

<
In another case, c<1 and d<1 and c<d, so that c<d if c, d are positive reals.>
If c & d are positive reals this means that c & d can be integers also.
What's wrong with my solution. I have proved that c is greater. However the answer says that d is greater.

Zulfi.
 
  • #7
zak100 said:
Hi,
Thanks for your help. Actually it says that " c & d are positive".

<Which is smaller 1/c or 1/d ?>
I think it depends on values. They can be equal.

<
In another case, c<1 and d<1 and c<d, so that c<d if c, d are positive reals.>
If c & d are positive reals this means that c & d can be integers also.
What's wrong with my solution. I have proved that c is greater. However the answer says that d is greater.

Zulfi.
Your last line of post #1 c=1+d is incorrect. Meanwhile, since 1/c=1+1/d, this says the right side of the equation is greater than 1. That means c must be less than 1 (to make the left side greater than 1) and therefore c can not be a positive integer.
 
  • #8
I guessed that the use of integer was a typo - the question did not make sense otherwise .

Now think about this :

If x is greater than y then is 1/x greater or less than 1/y ? Put some numbers in if it helps - say 10 for x and 5 for y .
 
  • #9
Hi,
Thanks for your response. I got my mistake. I am now able to understand book's solution.
According to the book,
c = d/ (d+1)
Since d is positive, so d+1> d. if d= 7 then 7/(7+1) (which is a fraction & less than d). So d > d/(d+1). But c=d/(d+1). So d>c.

Now your question:
"If x is greater than y then is 1/x greater or less than 1/y ? Put some numbers in if it helps - say 10 for x and 5 for y" .

i.e. x> y so 1/x < 1/y. Inequalities reverse. 10> 5 so 1/10 < 1/5.

Thanks for help.

Zulfi.
 
  • #11
zak100 said:
Hi,
I have a question with my original solution:
i.e
1/c = 1 + 1/d
Why can't we write:
c= 1 + d?
Because it is incorrect. See below.
zak100 said:
which is obtained by taking reciprocal of both sides.
The reciprocal of ##1 + \frac 1 d## is NOT 1 + d.
Here is an example using numbers.
$$\frac 1 {\frac 1 3} = 1 + \frac 1 {\frac 1 2}$$
Does it follow that ##\frac 1 3 = 1 + \frac 1 2##?
zak100 said:
 

1. How do you compare two fractions?

To compare two fractions, you can use the following steps:

  • Find the common denominator between the two fractions.
  • Convert both fractions to have the same denominator.
  • Compare the numerators of the fractions. The larger numerator indicates the larger fraction.

2. Can you compare fractions with different denominators?

Yes, you can compare fractions with different denominators by finding the common denominator between the two fractions and converting them to have the same denominator before comparing.

3. What is the difference between comparing fractions with like denominators and different denominators?

Comparing fractions with like denominators is simpler as the denominators are already the same, making it easier to compare the numerators. Comparing fractions with different denominators requires an extra step of finding the common denominator and converting the fractions before comparison.

4. How can you determine which fraction is larger when the numerators are the same?

If the numerators are the same, you can compare the denominators to determine which fraction is larger. The fraction with the smaller denominator is the larger fraction.

5. Can you compare more than two fractions at once?

Yes, you can compare more than two fractions at once by arranging them in order from least to greatest or greatest to least. You can also convert all the fractions to have the same denominator before comparison.

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