Can the quantity be written as 42 (skipping the in )? If yes, how would we know if 42 means 4 tens and 2 units or it means 4 units and 2 tenths?

Understand the Question In our place value system, the position of each digit determines its value relative to the units place. represents whole units and tenths (). The number represents tens and units (). A decimal point is necessary to separate whole units from fractional tenths so that place values are not confused. Step 1 · Express in Decimal Form Converting the mixed number into decimal form: In , is in the units place and is in the tenths place. Step 2 · Determine the Place Value of Digits in Writing in expanded form: In , is in the tens place and is in the units place. Step 3 · Compare the Values Comparing the two quantities: Therefore, cannot be written as . The number represents tens and units, whereas represents units and tenths. The decimal point is essential to distinguish between the two. Answer > No, cannot be written as . represents tens and units (), while represents units and tenths (). We use the decimal point to distinguish tenths from whole tens. Common Mistakes Ignoring the Decimal Separator: Assuming can represent units and tenths without a decimal point. In standard place value, always means . Place Value Shift: Skipping shifts the digit from the units place to the tens place, making the value times larger.

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Question 28

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

If yes, how would we know if 42 means 4 tens and 2 units or it means 4 units and 2 tenths?

Solution

Understand the Question

In our place value system, the position of each digit determines its value relative to the units place.
$4 \dfrac{2}{10}$ represents $4$ whole units and $2$ tenths ( $4 + 0.2 = 4.2$ ).
The number $42$ represents $4$ tens and $2$ units ( $40 + 2 = 42$ ).
A decimal point is necessary to separate whole units from fractional tenths so that place values are not confused.

Step 1 · Express $4 \dfrac{2}{10}$ in Decimal Form

Converting the mixed number into decimal form:

\begin{aligned} 4 \dfrac{2}{10} &= 4 + \dfrac{2}{10} \\[0.6em] &= 4 + 0.2 \\[0.6em] &= 4.2 \end{aligned}

In $4.2$ , $4$ is in the units place and $2$ is in the tenths place.

Step 2 · Determine the Place Value of Digits in $42$

Writing $42$ in expanded form:

\begin{aligned} 42 &= 4 \times 10 + 2 \times 1 \\[0.6em] &= 40 + 2 \end{aligned}

In $42$ , $4$ is in the tens place and $2$ is in the units place.

Step 3 · Compare the Values

Comparing the two quantities: $4.2 \neq 42$

Therefore, $4 \dfrac{2}{10}$ cannot be written as $42$ . The number $42$ represents $4$ tens and $2$ units, whereas $4.2$ represents $4$ units and $2$ tenths. The decimal point is essential to distinguish between the two.

Answer

No, $4 \dfrac{2}{10}$ cannot be written as $42$ .

$42$ represents $4$ tens and $2$ units ( $40 + 2$ ), while $4.2$ represents $4$ units and $2$ tenths ( $4 + 0.2$ ). We use the decimal point to distinguish tenths from whole tens.

Common Mistakes

Ignoring the Decimal Separator: Assuming $42$ can represent $4$ units and $2$ tenths without a decimal point. In standard place value, $42$ always means $40 + 2$ .
Place Value Shift: Skipping $\dfrac{1}{10}$ shifts the digit $4$ from the units place to the tens place, making the value $10$ times larger.

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 $2 \times \dfrac{1}{10}$ )?