Fractions | IT

Question 2

By dividing the whole chikki into 6 equal parts in different ways, we get 16\dfrac{1}{6} chikki pieces of different shapes. Are they of the same size?

Question diagram 1
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Solution
Understand the Question
  • A fraction 16\dfrac{1}{6} represents one part when a whole object is divided into 66 equal parts.
  • Size of a flat piece refers to its area or amount of chikki, not its shape.
  • As long as the whole chikki is the same and divided into 66 parts of equal area, every 16\dfrac{1}{6} piece will have the exact same size, regardless of whether it is shaped as a rectangle, triangle, or any other shape.

Step 1 · First Division into Six Equal Parts

In the first method, one whole chikki is divided into 66 equal rectangular pieces.Diagram 1

Each rectangular piece represents 16\dfrac{1}{6} of the whole chikki.

Step 2 · Second Division into Six Equal Parts

In the second method, the same whole chikki is divided into 66 equal triangular pieces.Diagram 2

Each triangular piece also represents 16\dfrac{1}{6} of the whole chikki.

Step 3 · Compare the Sizes

Both methods divide the same whole chikki into 66 equal parts:

  • Each rectangular piece =16= \dfrac{1}{6} of the whole chikki.
  • Each triangular piece =16= \dfrac{1}{6} of the whole chikki.

Since both are equal fractions of the exact same whole, their sizes (areas) are equal despite having different shapes.

Step 4 · Combine Fraction Pieces

Taking 22 pieces of size 16\dfrac{1}{6} gives:

16+16=26=2÷26÷2=13\begin{aligned} \dfrac{1}{6} + \dfrac{1}{6} &= \dfrac{2}{6} \\[0.6em] &= \dfrac{2 \div 2}{6 \div 2} \\[0.6em] &= \dfrac{1}{3} \end{aligned}

Thus, two 16\dfrac{1}{6} pieces make up 13\dfrac{1}{3} of the whole chikki.

Answer

Yes, they are of the same size (area), even though their shapes are different.

Common Mistakes
  • Confusing Shape with Size: Believing that different shapes (e.g. triangles vs rectangles) must have different sizes. Size refers to the area/amount of the whole, which is identical (16)\left(\dfrac{1}{6}\right) for both.
  • Comparing Different Wholes: 16\dfrac{1}{6} pieces are only guaranteed to be equal in size if they come from the same (or identical) whole.

More questions in IT

Q1

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.

Q2

By dividing the whole chikki into 6 equal parts in different ways, we get 16\dfrac{1}{6} chikki pieces of different shapes. Are they of the same size?

Q3

Do it once more! Fill in the blank boxes.

Q4

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

  1. 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.
Q5

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.

Q6

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

Q7

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

Now, let us classify these in two groups:

Q8

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

Q9

What do you observe?

  • Are the lengths 12\dfrac{1}{2} and 24\dfrac{2}{4} equal?
  • Are the lengths 24\dfrac{2}{4} and 48\dfrac{4}{8} equal?
Q10

Now, check whether 13\dfrac{1}{3} and 26\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 12\dfrac{1}{2} and 36\dfrac{3}{6} equal?

Q12

Answer the following questions after looking at the fraction wall:

Are 23\dfrac{2}{3} and 46\dfrac{4}{6} equivalent fractions? Why?

Q13

Answer the following questions after looking at the fraction wall:

How many pieces of length 16\dfrac{1}{6} will make a length of 12\dfrac{1}{2}?

Q14

Answer the following questions after looking at the fraction wall:

How many pieces of length 16\dfrac{1}{6} will make a length of 13\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 12\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

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 15<25\dfrac{1}{5} < \dfrac{2}{5}, 37<47\dfrac{3}{7} < \dfrac{4}{7}, and 12<58\dfrac{1}{2} < \dfrac{5}{8}.

Q20

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

  1. Group 1: 3 glasses of sugarcane juice divided equally among 4 children. Group 2: 7 glasses of sugarcane juice divided equally among 10 children.
  2. 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. 72\dfrac{7}{2} and 35\dfrac{3}{5}

b. 83\dfrac{8}{3} and 56\dfrac{5}{6}

c. 34\dfrac{3}{4} and 35\dfrac{3}{5}

d. 67\dfrac{6}{7} and 85\dfrac{8}{5}

e. 94\dfrac{9}{4} and 52\dfrac{5}{2}

f. 110\dfrac{1}{10} and 29\dfrac{2}{9}

g. 83\dfrac{8}{3} and 114\dfrac{11}{4}

h. 136\dfrac{13}{6} and 19\dfrac{1}{9}

Q22

Context: Meena's father made some chikki. Meena ate 12\dfrac{1}{2} of it and her younger brother ate 14\dfrac{1}{4} of it.

Q. How much of the total chikki is remaining?

Q23

Try adding 47+67\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!

  1. 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.

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