A Peek Beyond the PointClass 7 Mathematics NCERT Solutions

Answer: The scale with smaller divisions (marked in millimeters or tenths of a centimeter) helped to measure the length of the screws accurately.

Reason: The screws have lengths that are not exact whole numbers of centimeters. The smaller divisions allow for measuring the fractional parts of a centimeter, providing a more precise and accurate measurement. A scale with only centimeter markings would not be able to measure the slight difference in lengths between the two screws.

Answer: The measurement $2 \frac{7}{10} \mathrm{~cm}$ means the length of the screw is 2 full centimeters plus an additional seven-tenths of a centimeter.

Explanation: On a ruler, the unit length between two consecutive centimeter marks (e.g., between 2 and 3) is divided into 10 equal smaller parts. Each small part represents $\frac{1}{10}$ of a centimeter. To measure $2 \frac{7}{10} \mathrm{~cm}$, we count 2 whole centimeters from 0, and then count 7 of these smaller parts.

Answer: The unit (centimeter) was divided into smaller parts to achieve greater accuracy in measurement.

Explanation: The lengths of the screws were not exact whole centimeters. To measure the portion of the length that is less than a full centimeter, the centimeter unit needs to be subdivided. Dividing it into 10 equal parts (tenths) allows for a precise measurement of lengths that fall between the whole number marks.

Description of Objects: The picture shows three objects (a crayon, a key, and a safety pin) placed against a centimeter ruler, with one end at the 0 mark.

Description of Objects: The images show a carrot, a feather, and a toothbrush measured against scales marked in units and tenths of a unit.

Given: A list of lengths in fractional form.

To Find: Arrange the lengths in increasing order (from smallest to largest).

Solution: To compare the lengths, we first convert all mixed fractions to improper fractions with a denominator of 10. (a) $\frac{9}{10}$ (b) $1 \frac{7}{10} = \frac{10+7}{10} = \frac{17}{10}$ (c) $\frac{130}{10}$ (d) $13 \frac{1}{10} = \frac{130+1}{10} = \frac{131}{10}$ (e) $10 \frac{5}{10} = \frac{100+5}{10} = \frac{105}{10}$ (f) $7 \frac{6}{10} = \frac{70+6}{10} = \frac{76}{10}$ (g) $6 \frac{7}{10} = \frac{60+7}{10} = \frac{67}{10}$ (h) $\frac{4}{10}$

Now we compare the numerators: 9, 17, 130, 131, 105, 76, 67, 4. Arranging the numerators in increasing order: 4 < 9 < 17 < 67 < 76 < 105 < 130 < 131.

Therefore, the lengths in increasing order are: $\frac{4}{10} < \frac{9}{10} < \frac{17}{10} < \frac{67}{10} < \frac{76}{10} < \frac{105}{10} < \frac{130}{10} < \frac{131}{10}$

Writing them in their original form: $\frac{4}{10} < \frac{9}{10} < 1 \frac{7}{10} < 6 \frac{7}{10} < 7 \frac{6}{10} < 10 \frac{5}{10} < \frac{130}{10} < 13 \frac{1}{10}$

Final Answer: The lengths in increasing order are: (h) $\frac{4}{10}$, (a) $\frac{9}{10}$, (b) $1 \frac{7}{10}$, (g) $6 \frac{7}{10}$, (f) $7 \frac{6}{10}$, (e) $10 \frac{5}{10}$, (c) $\frac{130}{10}$, (d) $13 \frac{1}{10}$.

Given: The lengths $4 \frac{1}{10}, \frac{4}{10}, \frac{41}{10}, 41 \frac{1}{10}$.

To Find: Arrange the lengths in increasing order.

Solution: First, let's convert all numbers to a common format, either improper fractions or mixed numbers, to make comparison easier. Converting to improper fractions with a denominator of 10:

We observe that two of the lengths are the same: $4 \frac{1}{10} = \frac{41}{10}$.

Now, we compare the numerators of the fractions: 4, 41, 411. In increasing order, the numerators are 4 < 41 < 411.

So, the fractions in increasing order are: $\frac{4}{10} < \frac{41}{10} < \frac{411}{10}$

Writing them in their original forms: $\frac{4}{10} < 4 \frac{1}{10} = \frac{41}{10} < 41 \frac{1}{10}$

Final Answer: The lengths in increasing order are: $\frac{4}{10}, 4 \frac{1}{10}$ (or $\frac{41}{10}$), $41 \frac{1}{10}$.

Given: Length of lower arm = $2 \frac{7}{10}$ units Length of upper arm = $3 \frac{6}{10}$ units

To Find: The total length of Sonu's arm.

Solution: To find the total length, we need to add the lengths of the lower and upper arm. Total length = $2 \frac{7}{10} + 3 \frac{6}{10}$

Method 1: Adding whole parts and fractional parts separately $(2 + 3) + (\frac{7}{10} + \frac{6}{10})$ $= 5 + \frac{7+6}{10}$ $= 5 + \frac{13}{10}$ Since $\frac{13}{10}$ is an improper fraction, we convert it to a mixed number: $\frac{13}{10} = 1 \frac{3}{10}$. $= 5 + 1 \frac{3}{10}$ $= 6 \frac{3}{10} \text{ units}$

Method 2: Converting to improper fractions first $2 \frac{7}{10} = \frac{20+7}{10} = \frac{27}{10}$ $3 \frac{6}{10} = \frac{30+6}{10} = \frac{36}{10}$ Total length = $\frac{27}{10} + \frac{36}{10} = \frac{27+36}{10} = \frac{63}{10}$ Converting back to a mixed number: $\frac{63}{10} = 6 \frac{3}{10}$ units.

Final Answer: The total length of Sonu's arm is $6 \frac{3}{10}$ units.

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

To Find: The total length of the honeybee.

Solution: To find the total length, we add the lengths of all three parts. Total length = $2 \frac{3}{10} + 5 \frac{4}{10} + 7 \frac{5}{10}$

First, add the whole number parts: $2 + 5 + 7 = 14$

Next, add the fractional parts: $\frac{3}{10} + \frac{4}{10} + \frac{5}{10} = \frac{3+4+5}{10} = \frac{12}{10}$

Now, add the two sums together: Total length = $14 + \frac{12}{10}$ Convert the improper fraction $\frac{12}{10}$ to a mixed number: $\frac{12}{10} = 1 \frac{2}{10}$. Total length = $14 + 1 \frac{2}{10} = 15 \frac{2}{10}$ units.

Final Answer: The total length of the honeybee is $15 \frac{2}{10}$ units.

Given: Total length of hand = $12 \frac{4}{10}$ units Length of palm = $6 \frac{7}{10}$ units

To Find: The length of the longest (middle) finger.

Solution: The length of the finger is the difference between the total hand length and the palm length. Length of finger = $12 \frac{4}{10} - 6 \frac{7}{10}$

We need to subtract. Notice that the fraction in the first number ($\frac{4}{10}$) is smaller than the fraction in the second number ($\frac{7}{10}$). So, we need to borrow from the whole number part.

We can rewrite $12 \frac{4}{10}$ as $11 + 1 + \frac{4}{10}$. Since $1 = \frac{10}{10}$, we have: $11 + \frac{10}{10} + \frac{4}{10} = 11 + \frac{14}{10} = 11 \frac{14}{10}$.

Now the subtraction becomes: $11 \frac{14}{10} - 6 \frac{7}{10}$

Subtract the whole parts: $11 - 6 = 5$. Subtract the fractional parts: $\frac{14}{10} - \frac{7}{10} = \frac{7}{10}$.

Combine the results: $5 \frac{7}{10}$ units.

Method 2: Converting to improper fractions $12 \frac{4}{10} = \frac{124}{10}$ $6 \frac{7}{10} = \frac{67}{10}$ Length of finger = $\frac{124}{10} - \frac{67}{10} = \frac{124 - 67}{10} = \frac{57}{10}$ Converting back to a mixed number: $\frac{57}{10} = 5 \frac{7}{10}$ units.

Final Answer: The length of the longest (middle) finger is $5 \frac{7}{10}$ units.

Given: Length of Celestial Pearl Danio = $2 \frac{4}{10} \mathrm{~cm}$ Length of Philippine Goby = $\frac{9}{10} \mathrm{~cm}$

To Find: The difference in their lengths.

Solution: We need to calculate the difference: $2 \frac{4}{10} - \frac{9}{10}$. To perform the subtraction, we first convert the mixed number to an improper fraction. $2 \frac{4}{10} = \frac{2 \times 10 + 4}{10} = \frac{24}{10}$.

Now, subtract the fractions: Difference = $\frac{24}{10} - \frac{9}{10} = \frac{24 - 9}{10} = \frac{15}{10}$.

The difference is $\frac{15}{10} \mathrm{~cm}$. We can write this as a mixed number: $\frac{15}{10} = 1 \frac{5}{10} \mathrm{~cm}$. This can be simplified to $1 \frac{1}{2} \mathrm{~cm}$.

Final Answer: The difference in their lengths is $1 \frac{5}{10} \mathrm{~cm}$ (or $1.5 \mathrm{~cm}$).