What is a Benchmark?
A benchmark is a number or point of reference commonly used to compare other numbers. To compare fractions, \(\frac{1}{2}\) or 1 may be used as benchmarks.
The following are the models that are used to compare fractions:
- Fraction strips
- Number line
Fraction strips
In this method, strips of fractions of the same length are lined up. Then, depending on the fraction, each strip is divided into parts, such that each part of the strip represents the respective fraction.
After this, using a suitable benchmark, each part is compared and it is determined whether one of the fractions is equal to, greater than, or less than the other fraction by comparing the strips of the two fractions.
Example:
Compare \(\frac{2}{5} \) and \(\frac{3}{4}\)
Solution:
We take the benchmark as \(\frac{1}{2}\) as \(\frac{1}{2}\) lies between \(\frac{2}{5}\) and \(\frac{3}{4}\).
2 is less than half of 5 and \(\frac{2}{5}\) is less than \(\frac{1}{2}\).
\(2<\frac{1}{2}\times5\)
\(\frac{2}{5}<\frac{1}{2}\)
From the fraction strips, we can see that three-fourths, that is, (3 of \(\frac{1}{4}\) th)is greater than \(\frac{1}{2}\).
Since \(\frac{2}{5}<\frac{1}{2}\) and \(\frac{3}{4}>\frac{1}{2}\)
So, \(\frac{2}{5}<\frac{3}{4}\).
Number line
Fractions are plotted on an open number line, a benchmark is assumed, and each fraction is compared with the benchmark.
Based on the above comparison, both fractions are compared with each other to give a true statement if it is <, =, >.
On a number line, numbers that lie to the right of a particular number are greater than that number. Similarly, numbers that lie to the left of a given number are less than the given number.
For example:
Use a number line to compare \(\frac{6}{10}\) and \(\frac{2}{5}\).
Solution:
The greatest denominator in the two fractions is 10. So we draw a number line and mark 0 and 1. Then we create 10 divisions such that each division is \(\frac{1}{10},\frac{2}{10},\ \frac{3}{10}…\)
Post this, we mark the benchmark \(\frac{1}{2}\).
In the number line, \(\frac{1}{2}\) is represented as \(\frac{5}{10}\).
Now we plot \(\frac{6}{10}\).
\(\frac{6}{10}>\frac{1}{2}\) , as \(\frac{6}{10}\) lies to the right of \(\frac{1}{2}\).
Now we mark \(\frac{2}{5}\), which is represented as \(\frac{4}{10}\).
\(\frac{2}{5}<\frac{1}{2}\), as \(\frac{2}{5}\) lies to the left of \(\frac{1}{2}\).
Since \(\frac{6}{10}>\frac{1}{2}\) and \(\frac{2}{5}<\frac{1}{2}\)
So, \(\frac{6}{10}>\frac{2}{5}.\)
Finding Methods to Compare Fractions without using Benchmarks are
- Using like numerators
- Using like denominators
Using like numerators:
In this method, the numerators of both fractions are made equal by multiplying the numerators and denominators of the fraction by the same number.
For example:
Compare \(\frac{10}{12}\) and \(\frac{5}{7}\).
Step 1: Multiply the numerator and denominator of \(\frac{5}{7}\) by 2.
\(\frac{5}{7}\times\frac{2}{2}=\frac{10}{14}\)
Step 2: Compare the two fractions.
Here the whole parts are divided into two parts, 12 and 14. In \(\frac{10}{12}\) 10 out of the 12 parts are considered, but in \(\frac{10}{14}\) only 10 out of the 14 parts are considered. Take a look at the model below.
Though the number of parts is the same, \(\frac{10}{12}>\frac{10}{14}\), as more parts are taken into account when comparing with the whole in \(\frac{10}{12}\) than in \(\frac{10}{14}\).
So, \(\frac{10}{12}>\frac{5}{7}\)
Using like denominators:
In this method, the denominators of both the fractions are made the same by multiplying the numerator and the denominator by a suitable number. Based on this, the two fractions are compared.
For example:
Compare \(\frac{2}{3}\) and \(\frac{3}{4}\).
Solution:
Step 1: Multiply the numerator and the denominator of the fraction \(\frac{2}{3}\) by 4.
\(\frac{2}{3}\times\frac{4}{4}=\frac{8}{12}\)
Step 2: Multiply the numerator and the denominator of the fraction \(\frac{3}{4}\) by 3.
\(\frac{3}{4}\times\frac{3}{3}=\frac{9}{12}\)
Step 3: Compare the two fractions.
Here the wholes are divided into the same number of parts, 12. Hence, the comparison becomes easier. We know that 8 parts are less than 9 parts, as 8 < 9.
Hence, \(\frac{8}{12}<\frac{9}{12}\).
So,
\(\frac{2}{3}<\frac{3}{4}\)
Solved Compare Fraction Examples
Example 1:
Noah read a milkshake recipe:
Milkshake recipe
Milk \(\rightarrow\frac{1}{2}\) cup
Banana \(\rightarrow\frac{1}{2}\) cup
Noah has \(\frac{2}{4}\) cups of banana and \(\frac{3}{4}\) cups of milk. Does he have enough ingredients to make the milkshake?
Solution:
Step 1: Compare the quantity of ingredients on both sides.
Let’s convert the fractions into fractions that have the same denominator, 4.
Milk needed is \(\frac{1}{2}=\frac{1\times2}{2\times2}=\frac{2}{4}\) cup
Noah has \(\frac{3}{4}\) cups of milk.
3 > 2, so \(\frac{2}{4}<\frac{3}{4}\)
Therefore, \(\frac{1}{2}<\frac{3}{4}\).
So, he has enough milk. Let’s check the bananas.
Noah needs \(\frac{1}{2}\) cup bananas. He has \(\frac{2}{4}\) cups of bananas.
But \(\frac{1}{2}=\frac{2}{4}\). Hence, he has enough bananas as well.
So, Noah has enough ingredients to make the milkshake.
Example 2:
Aubrey baked a cake. She knew the cake would taste good, so she invited some of her friends over. Jacob, Natalie, and Katie came over and started eating the cake. They ate the cake so quickly that Aubrey did not get even a slice. The portion of the cake eaten by each is given below. Katie says that they all had the same quantity of cake. Is it true? If not, put the list of the friends in order, starting from those who ate the most to the least.
Friends | Weight(in tons) |
|---|---|
Katie | \(\frac{1}{8}\) |
Jacob | \(\frac{1}{2}\) |
Natalie | \(\frac{3}{8}\) |
Solution:
Step 1: Multiply the numerator and the denominator of the fraction \(\frac{1}{2}\) by 4
\(\frac{1}{2}\times\frac{4}{4}=\frac{4}{8}\)
Step 2: Since all three fractions have the same denominator, compare the 3 fractions.
We know that 4 > 3 > 1
So, \(\frac{4}{8}>\frac{3}{8}>\frac{1}{8}\)
Jacob > Natalie > Katie
Hence, Katie was not right; the cake was not evenly distributed.
If the denominators of two fractions are the same, the fraction with the greater numerator is greater. The fraction with a smaller numerator is a smaller fraction. Also, if the numerators are equal, then the fractions are equivalent.
Fractions are a part of a whole. If we have a strip or bar, we can consider it to be a whole. On dividing it into equal parts, each part can represent a fraction. The bar or strip is divided according to the denominator of the fraction.