A Peek Beyond the Point | IT

Question 19

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

Solution

Understand the Question

A ruler divides whole units into tenths (longer marks, where each represents $\dfrac{1}{10}$ ) and hundredths (smaller subdivisions between tenths, where each represents $\dfrac{1}{100}$ ).
Since $\dfrac{1}{10} = \dfrac{10}{100}$ , tenths can be converted to hundredths by multiplying both numerator and denominator by $10$ .
Measurements can be expressed as a sum of parts, a mixed fraction, an improper fraction, or read out in words based on place value.

(i) Write the measurement and read out in words for the first length: $5$ units, $3$ tenths, and $7$ hundredths.

Step 1 · Calculate and Express the First Measurement

Diagram 1

The length is $5$ whole units, $3$ tenths, and $7$ hundredths:

$5 + \dfrac{3}{10} + \dfrac{7}{100}$

Convert tenths to hundredths: $\dfrac{3}{10} = \dfrac{3 \times 10}{10 \times 10} = \dfrac{30}{100}$

\begin{aligned} 5 + \dfrac{30}{100} + \dfrac{7}{100} &= 5 \dfrac{37}{100} \end{aligned}

As an improper fraction:

\begin{aligned} 5 \dfrac{37}{100} &= \dfrac{5 \times 100 + 37}{100} \\[0.6em] &= \dfrac{537}{100} \end{aligned}

In words:

Five and three-tenths and seven-hundredths
Five and thirty-seven-hundredths
Five hundred thirty-seven-hundredths

Answer

(i) $5 \dfrac{37}{100}$ (or $\dfrac{537}{100}$ )

(ii) Write the measurement and read out in words for the second length: $15$ units and $3$ hundredths.

Step 1 · Calculate and Express the Second Measurement

The length is $15$ whole units and $3$ hundredths:

\begin{aligned} 15 + \dfrac{3}{100} &= 15 \dfrac{3}{100} \end{aligned}

As an improper fraction:

\begin{aligned} 15 \dfrac{3}{100} &= \dfrac{15 \times 100 + 3}{100} \\[0.6em] &= \dfrac{1503}{100} \end{aligned}

In words:

Fifteen and three-hundredths
One thousand five hundred three-hundredths

Answer

(ii) $15 \dfrac{3}{100}$ (or $\dfrac{1503}{100}$ )

(iii) Write the measurement and read out in words: $7 \dfrac{5}{10} \dfrac{2}{100}$ .

Step 1 · Calculate and Express the Third Measurement

The measurement represents $7$ whole units, $5$ tenths, and $2$ hundredths:

Convert tenths to hundredths ( $\dfrac{5}{10} = \dfrac{50}{100}$ ):

\begin{aligned} 7 + \dfrac{50}{100} + \dfrac{2}{100} &= 7 \dfrac{52}{100} \end{aligned}

As an improper fraction:

\begin{aligned} 7 \dfrac{52}{100} &= \dfrac{7 \times 100 + 52}{100} \\[0.6em] &= \dfrac{752}{100} \end{aligned}

In words:

Seven and five-tenths and two-hundredths
Seven and fifty-two-hundredths
Seven hundred fifty-two-hundredths

Answer

(iii) $7 \dfrac{52}{100}$ (or $\dfrac{752}{100}$ )

(iv) Write the measurement and read out in words: $9 \dfrac{8}{10}$ .

Step 1 · Calculate and Express the Fourth Measurement

The measurement represents $9$ whole units and $8$ tenths:

Convert tenths to hundredths: $\dfrac{8}{10} = \dfrac{8 \times 10}{10 \times 10} = \dfrac{80}{100}$

\begin{aligned} 9 + \dfrac{80}{100} &= 9 \dfrac{80}{100} \end{aligned}

As an improper fraction:

\begin{aligned} 9 \dfrac{80}{100} &= \dfrac{9 \times 100 + 80}{100} \\[0.6em] &= \dfrac{980}{100} \end{aligned}

In words:

Nine and eight-tenths
Nine and eighty-hundredths
Nine hundred eighty-hundredths

Answer

(iv) $9 \dfrac{8}{10}$ (or $9 \dfrac{80}{100}$ , or $\dfrac{980}{100}$ )

Common Mistakes

Place Value Confusion: Confusing tenths ( $\dfrac{1}{10}$ ) with hundredths ( $\dfrac{1}{100}$ ). Remember that $1 \text{ tenth} = 10 \text{ hundredths}$ .
Improper Fraction Conversion: When converting $5\dfrac{37}{100}$ , multiplying $5 \times 10$ instead of $5 \times 100$ before adding the numerator.

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 $2 \dfrac{7}{10} \text{ cm}$ (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 $2\dfrac{7}{10}\text{ cm}$ , we go from 0 to 2 and then take seven parts of $\dfrac{1}{10}$ . The length of the screw is $2\text{ cm}$ and $\dfrac{7}{10}\text{ cm}$ . Similarly, we can make sense of the length $3\dfrac{2}{10}\text{ cm}$ .

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) $\dfrac{9}{10}$

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

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

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

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

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

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

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

Q10

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

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

Q11

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

Q12

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

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

Q13

A Celestial Pearl Danio's length is $2\dfrac{4}{10} \text{ cm}$ , and the length of a Philippine Goby is $\dfrac{9}{10} \text{ cm}$ . 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) $4$ , $4\dfrac{3}{10}$ , $4\dfrac{6}{10}$ , ______, ______, ______

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

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

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

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

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) $\dfrac{3}{10}, \dfrac{3}{100}, \dfrac{33}{100}$

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

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

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

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

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

Q21

Context: What will be the sum of $15\ \dfrac{3}{10}\ \dfrac{4}{100}$ and $2\ \dfrac{6}{10}\ \dfrac{8}{100}$ ? 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?

\begin{aligned} &(400 + 80 + 3) + (200 + 60 + 8) \\ &= (400 + 200) + (80 + 60) + (3 + 8) \\ &= 600 + 140 + 11 \\ &= 600 + 150 + 1 \\ &= 700 + 50 + 1 \\ &= 751 \end{aligned}

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