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Thought-Process to Discover Knowledge

Welcome to nubtrek.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue
mathsFractionsArithmetics with Fractions

Comparing Fractions

Comparing two fractions of different place values or denominators is explained.

Comparison in first principles -- matching two quantities to find one as smaller, equal, or larger than the other -- is extended for fractions (numbers of different place-values).

Based on this a simplified procedure (find LCM of denominators and convert to like fractions) is explained.



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Let us quickly review comparing whole numbers and set on to understand comparing fractions.

Given numbers `7` and `3`, which number is larger?

  • `7`
  • `7`
  • `3`

The answer is '`7`'

Given numbers `4` and `6`, which number is smaller?

  • `6`
  • `4`
  • `4`

The answer is '`4`'

Given numbers `5` and `5`, which number is smaller?

  • One of them has to be smaller
  • The numbers are equal
  • The numbers are equal

The answer is 'The numbers are equal'

In whole numbers, we had studied the following.

Trichotomy Property of Comparison : Two numbers can be compared to find one of them as trichotomy property

 •  smaller or less

 •  equal

 •  greater or larger

Comparison by First Principle: Two quantities are matched one-to-one and compared in the count or magnitude of the quantities. As a result one of them is smaller or equal to or larger to the other. whole number comparison Example: Comparing the numbers `7` and `8`. The quantities represented by them is compared in the figure. It is found that `7` is smaller than `8`.

Simplified Procedure -- Comparison by ordered-sequence: To find if one numbers is larger or smaller than another number, the numbers are compared using the order `0,1,2,3,4,5,6,7,8,9, 10, cdots`. comparing on number line Example: Comparing the numbers `4` and `7`.

`4` is on the left-side to `7` in the order `0,1,2,3,4,5,6,7,8,9` and so `4` is smaller than `7`.

Comparison by Place-value -- Simplified Procedure to Compare Large Numbers: To find if one number is larger or smaller than another number, the digits at the highest place value are compared and if they are equal, then the digits at next lower place value are compared. comparing numbers simplified procedure

Given two integers or whole numbers, they can be compared as

 •  greater or larger

 •  smaller or lesser

 •  equal.

Can such comparison be done on two fractions?

Given two fractions `1/8` and `1/12`. comparing 1/8 and 1/12 The figure shows the fractions. `1/8` is shown in brown, and `1/12` is shown in purple color. Which fraction is larger?

  • `1/8`
  • `1/8`
  • `1/12`

The answer is '`1/8`'. From the picture it is clear that `1/8` is larger.

Given two fractions `3/4` and `5/8`. Which fraction is larger?

  • comparing numerators `3` and `5`, it is clear that `5/8` is larger
  • considering the place values are different, the fractions cannot be compared without converting them to like fractions
  • considering the place values are different, the fractions cannot be compared without converting them to like fractions

The answer is 'considering the place values are different, the fractions cannot be compared without converting them to like fractions'.

Given two fractions `3/4` and `5/8`. comparing 3/4 and 5/8 The figure shows the fractions. `3/4` is shown in brown, and `5/8` is shown in purple color. Which of the following helps to compare these two fractions?

  • Convert the fractions to like fractions having same place value
  • Convert the fractions to like fractions having same place value
  • Convert the fractions to the simplest form

The answer is 'Convert the fractions to like fractions having same place value '.

Given two fractions `3/4` and `5/8`. The fraction `3/4` is converted to `6/8` to make them like fractions.comparing 3/4 and 5/8 Both `6/8` and `5/8` have the same place value `1/8` (ie: same denominator `8`).

In this form, the numerators can be compared and `6/8` is larger, which means its equivalent fraction `3/4` is larger than `5/8`.

Two fractions can be compared by numerators after converting them into like fractions.

In whole numbers, `2` tens and `8` units are compared by converting `2` tens into `20` units. `20` units is larger than `8` units.

In fractions, the denominators represent the place values, and numbers of same place value can be compared. Numerators of like fractions are compared as whole numbers, and unlike fractions are converted to like fractions before the comparison.

Comparing fractions: First principles The numerators of two like fractions can be compared whole numbers.

Unlike fractions are first converted into like fractions to compare them.

Solved Exercise Problem:

Given `3/4` and `2/3`, Which fraction is smaller?comparing 3/4 and 2/3

  • `3/4`
  • `2/3`
  • `2/3`

The answer is '`2/3`'. First, convert them to like fractions

`text(LCM)(4,3) = 12`

`3/4 = 9/12`

`2/3 = 8/12`

Comparing the numerators `9` and `8`, it is concluded that `2/3` is smaller.

So far only positive fractions were considered for comparison. Fractions are directed numbers too. Fractions can be either positive or negative. Let us see how to compare such fractions.

Comparison of Integers -- First Principles: : Comparison is in terms of the amount received.

Amount given is smaller than amount received, as comparison is by amount received.

Larger amount given is smaller than smaller amount given, as comparison is by the amount received.

Comparison of Integers -- Simplified Procedure:

sign-property of comparison
 •  +ve and +ve are compared as whole numbers.
 •  When comparing +ve and -ve, the +ve value is larger irrespective of the absolute values of the numbers.
 •  When comparing -ve and -ve, the number with smaller absolute value is larger than the other.

The absolute values are compared as the simplified procedure detailed in whole numbers comparison by place-value.

Which of the following is larger? `3/4` or `-6/7`

  • `3/4`
  • `3/4`
  • `-6/7`
  • both of them

The answer is "`3/4`".

`text(received:)3/4` is larger than `text(given:)6/7`. Negative fractions are smaller than positive fractions.

Which of the following is larger than the other? `-3` or `-6`.

comparison is in terms of received. and
`-3 = text(given:)3`
`-6 = text(given:)6`

  • `-3`
  • `-3`
  • `-6`
  • both of them

The answer is "`-3`".

In terms of amount received, `text(given:)3` is larger than `text(given:)6`.

Negative fraction with larger absolute value is smaller than negative fraction with smaller absolute value.

Comparison of Negative and Positive fractions: : Comparison is in terms of the amount received.

Amount given is smaller than amount received, as comparison is by amount received.

Larger amount given is smaller than smaller amount given, as comparison is by the amount received.

Simplified Procedure:

sign-property of comparison
 •  +ve and +ve are compared as larger absolute value is larger fraction in value.
 •  When comparing +ve and -ve, the +ve value is larger irrespective of the absolute values of the numbers.
 •  When comparing -ve and -ve, the number with smaller absolute value is larger than the other.

Solved Exercise Problem:

Which of the following is smaller than the other? `1/2` or `-2/3`

  • `1/2`
  • `-2/3`
  • `-2/3`
  • both of them

The answer is "`-2/3`." As per sign property of comparison, the negative number is smaller than the positive number.

Solved Exercise Problem:

Which of the following is smaller than the other? `-1/2` or `-3/4`

  • `-1/2`
  • `-3/4`
  • `-3/4`
  • both of them

The answer is "`-3/4`"

Both the numbers are negative. So, the larger amount in negative is smaller in values. Comparing the numbers without the sign and converting them to like fractions, `3/4` is larger in value and so `-3/4` is smaller than `-1/2`.

Solved Exercise Problem:

Which of the following is larger than the other? `-3/4` or `-3/4`

  • first `-3/4`
  • second `-3/4`
  • the numbers are equal
  • the numbers are equal

The answer is "the numbers are equal"

Solved Exercise Problem:

Which of the following is larger than the other? `3/4` or `-3/4`

  • `+3/4`
  • `+3/4`
  • `-3/4`
  • the numbers are equal

The answer is "`3/4`". As per sign property of comparison, the positive number is larger than the negative number.

Now we know that fractions can be compared based on the count or numerator.

In whole numbers, we learned that the ordinal-property of the numbers is defined by `0<1<2<3
In integers, we learned that the ordinal-property of the numbers is extended to `cdots <-3<-2<-1<0<1<2<3< cdots`.

Where do the fractions fit in the ordered sequence of integers?

  • ordinal property is extended to fractions
  • ordinal property is extended to fractions
  • ordinal property is not applicable to fractions

The answer is "ordinal property is extended to fractions".

To understand the ordinal property of fractions, let us first consider fractions of same denominator. The ordinal property is readily extended to `cdots <(-3)/5<(-2)/5<(-1)/5<0<1/5<2/5<3/5< cdots` Do these include the integer values?

  • Yes, the integers are in the positions `(-5)/5` or `5/5` or `10/5` etc.
  • Yes, the integers are in the positions `(-5)/5` or `5/5` or `10/5` etc.
  • No, these are just fractions

The answer is "Yes, the integers are in the positions `(-5)/5` or `5/5` or `10/5` etc.".

The ordinal property is best captured by the number-line.

 •  Number-line of whole numbers starts from `0` and extends in one direction. It consists of points at positions `0,1,2, cdots`.

 •  Number-line of integers extends in both the directions. It consists of points at positions `cdots, -2,-1,0,1,2, cdots`

 •  Number-line of fractions extends in both the directions. numberline of fractions Which of the following describe the number-line of fractions?

  • the number-line of fractions is the points at positions `cdots, -2,-1,0,1,2, cdots`
  • the number-line of fractions looks like a continuous line
  • the number-line of fractions looks like a continuous line

The answer is "the number-line of fractions looks like continuous line". Note: If the fractions of form `p/q` is considered, then the number-line is not strictly continuous, though fractions are placed extremely close on the number-line. We will learn about this in higher-classes.

Consider the number line of fractions.numberline of fractions Where is the fraction `1/2` located in the number-line?

  • `1/2` is not located in number-line
  • `1/2` is between points `0` and `1`
  • `1/2` is between points `0` and `1`

The answer is "`1/2` is between points `0` and `1`". The line-segment between `0` and `1` is split into `2` equal pieces. `1/2` is at the position of `1` piece as shown in the figure.

Consider the number line of fractions.numberline of fractions Where is the fraction `2 3/4` located in the number-line?

  • `2 3//4` is not located in number-line
  • `2 3//4` is between points `2` and `3` at `3`rd position of `4` pieces
  • `2 3//4` is between points `2` and `3` at `3`rd position of `4` pieces

The answer is "`2 3//4` is between points `2` and `3` at `3`rd position of `4` pieces". The line-segment between `2` and `3` is split into `4` equal pieces. `2 3//4` is at the position of `3`rd piece as shown in the figure.

Given several fractions, how to find the largest fraction among them?

  • Compare the numerators
  • compare the denominators
  • Convert the fractions to like fractions and compare numerators
  • Convert the fractions to like fractions and compare numerators

The answer is 'Convert the fractions to like fractions and compare numerators'.

What does the word 'ascend' mean? ascending meaning

  • to go up or climb
  • to go up or climb
  • to write a reason for

The answer is 'to go up or climb'.

What is the order called when fractions are ordered from the smallest to the largest?ascending meaning

  • Pronunciation : Say the answer once
    Spelling: Write the answer once

The answer is 'ascending order'.

What does the word 'descending' mean? descending meaning

  • to move down
  • to move down
  • to write a reason for

The answer is 'to move down'.

What is the order called when fractions are ordered from the largest to the smallest?descending meaning

  • Pronunciation : Say the answer once
    Spelling: Write the answer once

The answer is 'descending order'.

Given several fractions, how to arrange them in ascending order?

  • order them by numerators
  • order them by denominators
  • Convert the fractions to like fractions and order
  • Convert the fractions to like fractions and order

The answer is 'Convert the fractions to like fractions and order'.

Solved Exercise Problem:

Arrange the numbers in ascending order `-23/4`, `7/3`, `42/3`.

  • `7/3`, `-23/4`,`42/3`
  • `-23/4`, `7/3`, `42/3`
  • `-23/4`, `7/3`, `42/3`

The answer is "`-23/4`, `7/3`, `42/3`"

Solved Exercise Problem:

Arrange the numbers in descending order `8/5`, `9/10`, `-9/2`.

  • `-9/2`, `8/5`, `9/10`
  • `8/5`, `9/10`,`-9/2`
  • `8/5`, `9/10`,`-9/2`

The answer is "`8/5`, `9/10`,`-9/2`"

Two or more fractions can be compared to arrange them in
 •  ascending order : from the smallest to the largest
 •  descending order : from the largest to the smallest

Solved Exercise Problem:

Arrange the numbers in descending order `2/27`, `2/9`, `2/36`.

  • `2/36`, `2/27`, `2/9`
  • `2/9`, `2/27`, `2/36`
  • `2/9`, `2/27`, `2/36`

The answer is "`2/9`, `2/27`, `2/36`"

                            
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