A Peek Beyond the Point | IT

Question 21

Context: What will be the sum of $15\ \frac{3}{10}\ \frac{4}{100}$ and $2\ \frac{6}{10}\ \frac{8}{100}$ ? This can be solved in different ways. Some are shown below.

Are both these methods different?

Solution

Both methods add the numbers by combining like place values and regrouping when necessary.

Step 1 — Understand the problem

The problem asks us to find the sum of two numbers. The numbers are $15 + \frac{3}{10} + \frac{4}{100}$ and $2 + \frac{6}{10} + \frac{8}{100}$ . We need to determine if the two presented methods are different.

Step 2 — Analyze Method 1

Method 1 adds the whole numbers, tenths, and hundredths separately. First, it adds the whole parts. Then, it adds the tenths parts. Next, it adds the hundredths parts. The sum is $17 + \frac{9}{10} + \frac{12}{100}$ . It then regroups the hundredths. $\frac{12}{100}$ is equal to $\frac{10}{100} + \frac{2}{100}$ . We know that $\frac{10}{100}$ is the same as $\frac{1}{10}$ . So, $\frac{12}{100}$ becomes $\frac{1}{10} + \frac{2}{100}$ . The sum becomes $17 + \frac{9}{10} + \frac{1}{10} + \frac{2}{100}$ . Now, it combines the tenths. $\frac{9}{10} + \frac{1}{10} = \frac{10}{10}$ . We know that $\frac{10}{10}$ is equal to $1$ . So, the sum becomes $17 + 1 + \frac{2}{100}$ . Finally, it adds the whole numbers. $17 + 1 = 18$ . The final sum is $18 + \frac{2}{100}$ .

$(15 + 2) + \left(\frac{3}{10} + \frac{6}{10}\right) + \left(\frac{4}{100} + \frac{8}{100}\right)$

$= 17 + \frac{9}{10} + \frac{12}{100}$

$= 17 + \frac{9}{10} + \frac{10}{100} + \frac{2}{100}$

$= 17 + \frac{9}{10} + \frac{1}{10} + \frac{2}{100}$

$= 17 + \frac{10}{10} + \frac{2}{100}$

$= 17 + 1 + \frac{2}{100}$

$\boxed{18\ \frac{2}{100}}$

Diagram 2

Step 3 — Analyze Method 2

Method 2 uses a vertical column addition format. It aligns the whole numbers, tenths, and hundredths. First, it adds the hundredths column. $\frac{4}{100} + \frac{8}{100} = \frac{12}{100}$ . Next, it adds the tenths column. $\frac{3}{10} + \frac{6}{10} = \frac{9}{10}$ . Then, it adds the whole numbers column. $15 + 2 = 17$ . The intermediate sum is $17\ \frac{9}{10}\ \frac{12}{100}$ . It then regroups the hundredths. $\frac{12}{100}$ is $\frac{10}{100} + \frac{2}{100}$ . This means $\frac{1}{10} + \frac{2}{100}$ . The $\frac{1}{10}$ is carried over to the tenths column. So, the tenths column becomes $\frac{9}{10} + \frac{1}{10} = \frac{10}{10}$ . The hundredths column becomes $\frac{2}{100}$ . The intermediate sum is $17\ \frac{10}{10}\ \frac{2}{100}$ . It then regroups the tenths. $\frac{10}{10}$ is equal to $1$ . This $1$ is carried over to the whole numbers column. So, the whole numbers column becomes $17 + 1 = 18$ . The tenths column becomes $0$ . The final sum is $18\ \frac{2}{100}$ .

$\begin{array}{c c c c c c} & 15 & \frac{3}{10} & & \frac{4}{100} \\ + & 2 & \frac{6}{10} & & \frac{8}{100} \\ \hline \\ = & 17 & \frac{9}{10} & & \frac{12}{100} \\ \\ = & 17 & \frac{10}{10} & & \frac{2}{100} \\ \\ = & 18 & & & \frac{2}{100} \end{array}$

$\boxed{18\ \frac{2}{100}}$

<DIAGRAM: No diagram needed for this step.>

Step 4 — Compare the methods

Both methods perform the same mathematical operations. They both add the whole numbers, tenths, and hundredths. They both use the concept of regrouping or carrying over. Method 1 shows the regrouping steps horizontally. Method 2 shows the regrouping steps vertically. The final answer from both methods is the same.

Answer

(i) Yes, both these methods are different in their presentation and structure. (ii) Method 1 shows the addition horizontally, grouping like terms and then regrouping. (iii) Method 2 shows the addition vertically, like column addition, and implicitly carries over values.

More questions in IT

In the following figure, screws are placed above a scale. Measure them and write their length in the space provided.

Which scale helped you measure the length of the screws accurately? Why?

What is the meaning of $2 \frac{7}{10}$ cm (the length of the first screw)?

Context: As seen on the ruler, the unit length between two consecutive numbers is divided into 10 equal parts. To get the length $2\frac{7}{10}\text{ cm}$ , we go from 0 to 2 and then take seven parts of $\frac{1}{10}$ . The length of the screw is $2\text{ cm}$ and $\frac{7}{10}\text{ cm}$ . Similarly, we can make sense of the length $3\frac{2}{10}\text{ cm}$ .

Q. Can you explain why the unit was divided into smaller parts to measure the screws?

Measure the following objects using a scale and write their measurements in centimeters (as shown earlier for the lengths of the screws): pen, sharpener, and any other object of your choice.

Write the measurements of the objects shown in the picture:

For the objects shown below, write their lengths in two ways and read them aloud. An example is given for the USB cable. (Note that the unit length used in each diagram is not the same).

Arrange these lengths in increasing order:

(a) $\frac{9}{10}$

(b) $1 \frac{7}{10}$

(d) $13 \frac{1}{10}$

(e) $10 \frac{5}{10}$

(f) $7 \frac{6}{10}$

(g) $6 \frac{7}{10}$

(h) $\frac{4}{10}$

Arrange the following lengths in increasing order: $4\frac{1}{10}$ , $\frac{4}{10}$ , $\frac{41}{10}$ , $41\frac{1}{10}$ .

Q10

The lengths of the body parts of a honeybee are given. Find its total length.

Head: $2 \frac{3}{10}$ units Thorax: $5 \frac{4}{10}$ units Abdomen: $7 \frac{5}{10}$ units

Q11

The length of Shylaja's hand is $12 \frac{4}{10}$ units, and her palm is $6 \frac{7}{10}$ units, as shown in the picture. What is the length of the longest (middle) finger?

Q12

Context: The length of Shylaja's hand is $12 \frac{4}{10}$ units, and her palm is $6 \frac{7}{10}$ units.

Q. Try computing the difference by converting both lengths to tenths.

Q13

A Celestial Pearl Danio's length is $2 \frac{4}{10}$ cm, and the length of a Philippine Goby is $\frac{9}{10}$ cm. What is the difference in their lengths?

Q14

How big are these fish compared to your finger?

Q15

Observe the given sequences of numbers. Identify the change after each term and extend the pattern:

(a) $4$ , $4\frac{3}{10}$ , $4\frac{6}{10}$ , ______, ______, ______

(b) $8\frac{2}{10}$ , $8\frac{7}{10}$ , $9\frac{2}{10}$ , ______, ______, ______

(d) $5\frac{7}{10}$ , $5\frac{3}{10}$ , ______, ______, ______, ______

(e) $13\frac{5}{10}$ , $13$ , $12\frac{5}{10}$ , ______, ______, ______, ______

(f) $11\frac{5}{10}$ , $10\frac{4}{10}$ , $9\frac{3}{10}$ , ______, ______, ______

Q16

What is the length of this smaller part? How many such smaller parts make a unit length?

Q17

How many one-hundredths make one-tenth? Can we also say that the length is 4 units and 45 one-hundredths?

Q18

Observe the figure below. Notice the markings and the corresponding lengths written in the boxes when measured from 0. Fill the lengths in the empty boxes.

Q19

For the lengths shown below write the measurements and read out the measures in words.

Q20

In each group, identify the longest and the shortest lengths. Mark each length on the scale.

(a) $\frac{3}{10}, \frac{3}{100}, \frac{33}{100}$

(b) $3\frac{1}{10}, \frac{30}{10}, 1\frac{3}{10}$

(d) $3\frac{6}{10}, 3\frac{6}{100}, 3\frac{6}{10}\frac{6}{100}$

(e) $\frac{8}{10}, \frac{2}{100}, \frac{9}{100}, 1\frac{8}{100}$

(f) $7\frac{3}{10}\frac{5}{100}, 7\frac{5}{10}, 7\frac{41}{100}$

(g) $\frac{65}{10}\frac{15}{100}, 5\frac{87}{100}, 5\frac{7}{100}$

Q21

Context: What will be the sum of $15\ \frac{3}{10}\ \frac{4}{100}$ and $2\ \frac{6}{10}\ \frac{8}{100}$ ? This can be solved in different ways. Some are shown below.

Are both these methods different?

Q22

Observe the addition done below for $483 + 268$ . Do you see any similarities between the methods shown above?

$(400 + 80 + 3) + (200 + 60 + 8)$

$= (400 + 200) + (80 + 60) + (3 + 8)$

$= 600 + 140 + 11$

$= 600 + 150 + 1$

$= 700 + 50 + 1$

$= 751$

Q25

Can we extend this further?

Q26

We can ask similar questions about fractional parts:

(a) How many thousandths make one unit? (b) How many thousandths make one tenth? (c) How many thousandths make one hundredth? (d) How many tenths make one ten? (e) How many hundredths make one ten?

Q27

Make a few more questions of this kind and answer them.

Q28

Can the quantity $4 \frac{2}{10}$ be written as 42 (skipping the $\frac{1}{10}$ in $2 \times \frac{1}{10}$ )?