Fractions | IT

Question 19

Suppose the number of children is kept the same, but the number of units that are being shared is increased? What can you say about each child's share now? Why? Discuss how your reasoning explains $\dfrac{1}{5} < \dfrac{2}{5}$ , $\dfrac{3}{7} < \dfrac{4}{7}$ , and $\dfrac{1}{2} < \dfrac{5}{8}$ .

Solution

Understand the Question

A fraction $\dfrac{\text{Numerator}}{\text{Denominator}}$ can be understood as sharing a given number of units (the numerator) equally among a fixed number of children (the denominator).
When the number of children remains the same (same denominator) but the number of units being shared increases (larger numerator), each child receives a larger share.
When denominators are different, we first convert the fractions to have a common denominator (same number of children) before comparing.

Step 1 · Effect of Increasing Units on Each Share

When more units are shared among the same number of children, each child's share increases. Diagram 1

For example, sharing $1$ unit among $5$ children gives each child $\dfrac{1}{5}$ . Sharing $2$ units among the same $5$ children gives each child $\dfrac{2}{5}$ . Since there are more units to divide among the same group, $\dfrac{2}{5} > \dfrac{1}{5}$ .

Step 2 · Explain $\dfrac{1}{5} < \dfrac{2}{5}$

Both fractions have a denominator of $5$ , meaning $5$ children share the items:

$\dfrac{1}{5}$ represents sharing $1$ unit among $5$ children.
$\dfrac{2}{5}$ represents sharing $2$ units among $5$ children.

Sharing $2$ units gives more to each child than sharing $1$ unit. $\dfrac{1}{5} < \dfrac{2}{5}$

Step 3 · Explain $\dfrac{3}{7} < \dfrac{4}{7}$

Both fractions have a denominator of $7$ , meaning $7$ children share the items:

$\dfrac{3}{7}$ represents sharing $3$ units among $7$ children.
$\dfrac{4}{7}$ represents sharing $4$ units among $7$ children.

Sharing $4$ units gives more to each child than sharing $3$ units. $\dfrac{3}{7} < \dfrac{4}{7}$

Step 4 · Explain $\dfrac{1}{2} < \dfrac{5}{8}$

To compare fractions with different denominators, express $\dfrac{1}{2}$ with a denominator of $8$ :

\begin{aligned} \dfrac{1}{2} &= \dfrac{1 \times 4}{2 \times 4} \\[0.6em] &= \dfrac{4}{8} \end{aligned}

Now compare $\dfrac{4}{8}$ and $\dfrac{5}{8}$ :

$\dfrac{4}{8}$ represents sharing $4$ units among $8$ children.
$\dfrac{5}{8}$ represents sharing $5$ units among $8$ children.

Since sharing $5$ units gives each child a larger share than sharing $4$ units: $\dfrac{4}{8} < \dfrac{5}{8} \implies \dfrac{1}{2} < \dfrac{5}{8}$

Answer

Each child's share increases because more total quantity is distributed among the same number of people. Thus, like fractions with larger numerators are always greater: $\dfrac{1}{5} < \dfrac{2}{5}$ , $\dfrac{3}{7} < \dfrac{4}{7}$ , and $\dfrac{1}{2} = \dfrac{4}{8} < \dfrac{5}{8}$ .

Common Mistakes

Comparing Unlike Denominators Directly: Directly comparing numerators in $\dfrac{1}{2}$ and $\dfrac{5}{8}$ without first converting $\dfrac{1}{2}$ to an equivalent fraction $\dfrac{4}{8}$ .
Confusing Numerator and Denominator: Remembering that the numerator is the quantity shared (increasing it increases the share), whereas the denominator is the number of shares/children (increasing it decreases each share).

More questions in IT

Arrange these fraction words in order of size from the smallest to the biggest in the empty box below:

One and a half, three quarters, one and a quarter, half, quarter, two and a half.

By dividing the whole chikki into 6 equal parts in different ways, we get $\dfrac{1}{6}$ chikki pieces of different shapes. Are they of the same size?

Do it once more! Fill in the blank boxes.

Now, can you find the lengths of the various blue lines shown below? Fill in the boxes as well.

Here, the fractional unit is dividing a length of 1 unit into three equal parts. Write the fraction that gives the length of the blue line in the box or in your notebook.

Here, a unit is divided into 5 equal parts. Write the fraction that gives the length of the blue lines in the respective boxes or in your notebook.

Now, a unit is divided into 8 equal parts. Write the appropriate fractions in your notebook.

Write down all the fractions you marked on the number line earlier.

Now, let us classify these in two groups:

Did you notice something common between the fractions that are greater than 1?

What do you observe?

Are the lengths $\dfrac{1}{2}$ and $\dfrac{2}{4}$ equal?
Are the lengths $\dfrac{2}{4}$ and $\dfrac{4}{8}$ equal?

Q10

Now, check whether $\dfrac{1}{3}$ and $\dfrac{2}{6}$ are equivalent fractions or not, using paper strips.

Q11

Answer the following questions after looking at the fraction wall:

Are the lengths $\dfrac{1}{2}$ and $\dfrac{3}{6}$ equal?

Q12

Answer the following questions after looking at the fraction wall:

Are $\dfrac{2}{3}$ and $\dfrac{4}{6}$ equivalent fractions? Why?

Q13

Answer the following questions after looking at the fraction wall:

How many pieces of length $\dfrac{1}{6}$ will make a length of $\dfrac{1}{2}$ ?

Q14

Answer the following questions after looking at the fraction wall:

How many pieces of length $\dfrac{1}{6}$ will make a length of $\dfrac{1}{3}$ ?

Q15

Context: Anil was in a group where 2 cakes were divided equally among 5 children.

Q. Now, if there are 10 children in my group, how many cakes will I need so that they get same amount of cake as Anil?

Q16

What if we put two such groups together? One group where 2 cakes are divided equally between 5 children, and another group again with 4 cakes and 10 children.

Q17

Find some more fractions equivalent to $\dfrac{1}{2}$ . Write them in the boxes here:

Q18

Equally divide the rotis in the situations shown below and write down the share of each child. Are the shares in each of these cases the same? Why?

Q19

Q20

Now, decide in which of the two groups will each child get a larger share:

Group 1: 3 glasses of sugarcane juice divided equally among 4 children. Group 2: 7 glasses of sugarcane juice divided equally among 10 children.
Group 1: 4 glasses of sugarcane juice divided equally among 7 children. Group 2: 5 glasses of sugarcane juice divided equally among 7 children.

Which groups were easier to compare? Why?

Q21

Find equivalent fractions for the given pairs of fractions such that the fractional units are the same.

a. $\dfrac{7}{2}$ and $\dfrac{3}{5}$

b. $\dfrac{8}{3}$ and $\dfrac{5}{6}$

c. $\dfrac{3}{4}$ and $\dfrac{3}{5}$

d. $\dfrac{6}{7}$ and $\dfrac{8}{5}$

e. $\dfrac{9}{4}$ and $\dfrac{5}{2}$

f. $\dfrac{1}{10}$ and $\dfrac{2}{9}$

g. $\dfrac{8}{3}$ and $\dfrac{11}{4}$

h. $\dfrac{13}{6}$ and $\dfrac{1}{9}$

Q22

Context: Meena's father made some chikki. Meena ate $\dfrac{1}{2}$ of it and her younger brother ate $\dfrac{1}{4}$ of it.

Q. How much of the total chikki is remaining?

Q23

Try adding $\dfrac{4}{7} + \dfrac{6}{7}$ using a number line. Do you get the same answer?

Q24

Try doing this same exercise using the number line.

Q25

Puzzle!

Can you find three different fractional units that add up to 1?

It turns out there is only one solution to this problem (up to changing the order of the 3 fractions)! Can you find it? Try to find it before reading further.