Another Peek Beyond the Point — Class 7 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 7 Mathematics chapter "Another Peek Beyond the Point" — 7 important questions with detailed answers for CBSE board exam preparation.
TL;DR: Free step-by-step NCERT solutions for Class 7 Mathematics chapter "Another Peek Beyond the Point" — 7 important questions with detailed answers for CB…
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NCERT Solutions for Class 7 Mathematics — Another Peek Beyond the Point. Step-by-step answers to all 7 textbook questions from this chapter, written for CBSE board preparation and free to use.
Another Peek Beyond the Point — All 7 Questions Solved
Q1. Define decimal numbers and explain place value for 45.678.
Decimal Numbers:
Decimal numbers are numbers that contain a decimal point (.) separating the whole number part from the fractional part.
They are used to represent values between whole numbers with greater precision.
---
Place Value for 45.678:
Breakdown:
45.678 = 40 + 5 + 0.6 + 0.07 + 0.008
Place Value Chart:
| Digit | Place | Place Value | Value |
|-------|-------|-------------|-------|
| 4 | Tens | 10 | 40 |
| 5 | Ones/Units | 1 | 5 |
| . | Decimal Point | — | — |
| 6 | Tenths | 1/10 = 0.1 | 0.6 |
| 7 | Hundredths | 1/100 = 0.01 | 0.07 |
| 8 | Thousandths | 1/1000 = 0.001 | 0.008 |
---
Expanded Form:
45.678 = (4 × 10) + (5 × 1) + (6 × 0.1) + (7 × 0.01) + (8 × 0.001)
45.678 = 40 + 5 + 0.6 + 0.07 + 0.008
---
Reading the Number:
45.678 is read as "forty-five point six seven eight" or "forty-five and six hundred seventy-eight thousandths."
Key Points:
• Digits to the left of the decimal point represent whole numbers
• Digits to the right of the decimal point represent fractions (parts of 1)
• Each place to the right is 1/10 of the previous place
Q2. Multiply: 2.5 × 3.4. Show all steps including decimal placement.
Multiply: 2.5 × 3.4
Method: Multiply as whole numbers, then place the decimal point
Step 1: Ignore the decimal points and multiply as whole numbers
25 × 34
Multiplying 25 × 34:
25
× 34
----
100 (25 × 4)
750 (25 × 30)
----
850
Step 2: Count the decimal places in the original numbers
2.5 has 1 decimal place
3.4 has 1 decimal place
Total decimal places = 1 + 1 = 2
Step 3: Place the decimal point
We need 2 decimal places in the answer.
From 850, moving decimal 2 places to the left: 8.50 or 8.5
Step 4: Verify using fractions
2.5 = 25/10
3.4 = 34/10
(25/10) × (34/10) = (25 × 34)/(10 × 10) = 850/100 = 8.5 ✓
Answer: 2.5 × 3.4 = 8.5
---
Key Rule for Decimal Multiplication:
Total decimal places in the answer = Sum of decimal places in both numbers
Q3. Divide: 7.5 ÷ 2.5. Show the steps and verify the answer.
Divide: 7.5 ÷ 2.5
Method: Convert to whole number division by multiplying both by 10
Step 1: Count decimal places
7.5 has 1 decimal place
2.5 has 1 decimal place
Maximum decimal places = 1
Step 2: Multiply both numbers by 10 (to eliminate decimals)
7.5 × 10 = 75
2.5 × 10 = 25
Now we have: 75 ÷ 25
Step 3: Perform whole number division
75 ÷ 25 = 3
Verification:
25 × 3 = 75 ✓
Step 4: Find the answer
Since we multiplied both numbers by the same value, the quotient remains the same.
7.5 ÷ 2.5 = 3
Alternative Verification:
2.5 × 3 = 7.5 ✓
Answer: 7.5 ÷ 2.5 = 3
---
Key Rule for Decimal Division:
Multiply both divisor and dividend by 10, 100, 1000, etc., to make the divisor a whole number, then divide.
Q4. Add: 12.35 + 8.47 and subtract: 15.6 - 7.25. Align decimal points clearly.
Problem 1: Add 12.35 + 8.47
Step 1: Align decimal points vertically
12.35
+ 8.47
-------
Step 2: Add from right to left
Hundredths place: 5 + 7 = 12 (write 2, carry 1)
Tenths place: 3 + 4 + 1 = 8
Ones place: 2 + 8 = 10 (write 0, carry 1)
Tens place: 1 + 1 = 2
Step 3: Write the answer
12.35
+ 8.47
-------
20.82
Answer: 12.35 + 8.47 = 20.82
---
Problem 2: Subtract 15.6 - 7.25
Step 1: Align decimal points (add zeros if needed)
15.60
- 7.25
--------
Step 2: Subtract from right to left
Hundredths place: 0 - 5 (cannot do, borrow from tenths)
Borrow: 10 - 5 = 5
Tenths place: 6 - 1 (after borrowing) - 2 = 3
Ones place: 5 - 7 (cannot do, borrow from tens)
Borrow: 15 - 7 = 8
Tens place: 1 - 1 (after borrowing) = 0
Step 3: Write the answer
15.60
- 7.25
--------
8.35
Answer: 15.6 - 7.25 = 8.35
---
Key Rules:
• Always align decimal points vertically
• Add zeros after decimal if needed to match decimal places
• Follow the same addition/subtraction rules as whole numbers
Q5. Convert 0.625 to a fraction in simplest form.
Convert 0.625 to a Fraction:
Step 1: Count the decimal places
0.625 has 3 decimal places
Step 2: Write as a fraction with denominator as a power of 10
Number of decimal places = 3
Denominator = 10³ = 1000
Fraction = 625/1000
Step 3: Simplify by finding GCD
Find GCD of 625 and 1000:
Factors of 625: 1, 5, 25, 125, 625
Factors of 1000: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000
Common factors: 1, 5, 25, 125
GCD = 125
Step 4: Divide both numerator and denominator by GCD
625 ÷ 125 = 5
1000 ÷ 125 = 8
Simplified fraction = 5/8
Step 5: Verify
5 ÷ 8 = 0.625 ✓
Answer: 0.625 = 5/8
Q6. A length of 2.5 m is cut into 5 equal pieces. What is the length of each piece?
Given:
Total length = 2.5 m
Number of equal pieces = 5
Step 1: Set up the division
Length of each piece = Total length ÷ Number of pieces
Length of each piece = 2.5 ÷ 5
Step 2: Convert to whole number division
Multiply both by 10:
2.5 × 10 = 25
5 × 10 = 50
Now: 25 ÷ 50 (or simplify as 2.5 ÷ 5)
Alternative: Direct division
25 ÷ 5 = 5 (in tenths)
So 2.5 ÷ 5 = 0.5
Step 3: Perform the division
2.5 ÷ 5 = 0.5 m
Step 4: Verify
0.5 × 5 = 2.5 ✓
Answer: Each piece has a length of 0.5 m (or 50 cm).
Q7. The cost of 3.5 kg of rice is Rs. 245. What is the cost per kilogram?
Given:
Total cost = Rs. 245
Total weight = 3.5 kg
Step 1: Set up the division
Cost per kg = Total cost ÷ Total weight
Cost per kg = 245 ÷ 3.5
Step 2: Convert to whole number division
Multiply both by 10 to eliminate the decimal in the divisor:
245 × 10 = 2450
3.5 × 10 = 35
Now: 2450 ÷ 35
Step 3: Perform the long division
70
----
35)2450
245 (35 × 7 = 245)
----
0
2450 ÷ 35 = 70
Step 4: Verify
70 × 3.5 = 245 ✓
Verify: 70 × 3 = 210 and 70 × 0.5 = 35
Total = 210 + 35 = 245 ✓
Answer: The cost per kilogram of rice is Rs. 70.
---
Alternative Quick Method:
245 ÷ 3.5 = 2450 ÷ 35 = 245 ÷ 3.5
Simplify: (245 × 2) ÷ (3.5 × 2) = 490 ÷ 7 = 70
Another Peek Beyond the Point — Practice MCQs with Answers
Attempt these 20 multiple-choice questions after working through the solutions above. Each carries the correct option and the reasoning behind it, so a wrong answer tells you which idea to revisit.
MCQ 1. Which of the following best describes a point in geometry?
- A. A straight path extending infinitely in one direction.
- B. A location in space that has no size or dimension.
- C. A flat surface extending infinitely in all directions.
- D. A collection of points forming a boundary.
Answer: B. A location in space that has no size or dimension. — A point is a fundamental concept in geometry, representing an exact location in space without any dimension, size, or thickness.
MCQ 2. What is the sum of two complementary angles?
- A. 90 degrees
- B. 180 degrees
- C. 360 degrees
- D. 0 degrees
Answer: A. 90 degrees — Two angles are said to be complementary if their sum is exactly 90 degrees.
MCQ 3. If two lines intersect, how many points of intersection can they have?
- A. Zero
- B. One
- C. Two
- D. Infinitely many
Answer: B. One — Two distinct lines in a plane can intersect at most at one unique point. If they intersect at more than one point, they would be the same line.
MCQ 4. What kind of angle is formed by the hands of a clock at 3:00 PM?
- A. Acute angle
- B. Obtuse angle
- C. Right angle
- D. Straight angle
Answer: C. Right angle — At 3:00 PM, the minute hand is at 12 and the hour hand is at 3, forming an angle of 90 degrees, which is a right angle.
MCQ 5. Which term describes two angles that share a common vertex and a common arm but do not overlap?
- A. Complementary angles
- B. Supplementary angles
- C. Vertically opposite angles
- D. Adjacent angles
Answer: D. Adjacent angles — Adjacent angles are angles that are next to each other, sharing a common vertex and one common arm, without any interior points in common.
MCQ 6. If an angle measures 45 degrees, what is the measure of its complement?
- A. 135 degrees
- B. 45 degrees
- C. 90 degrees
- D. 180 degrees
Answer: B. 45 degrees — Complementary angles sum to 90 degrees. So, 90 - 45 = 45 degrees.
MCQ 7. What is a line segment?
- A. A line that extends infinitely in both directions.
- B. A part of a line with two distinct endpoints.
- C. A part of a line with one endpoint and extending infinitely in one direction.
- D. A curved path between two points.
Answer: B. A part of a line with two distinct endpoints. — A line segment is a finite portion of a line that is defined by two endpoints and includes all points on the line between them.
MCQ 8. If two angles form a linear pair, what is their sum?
- A. 90 degrees
- B. 180 degrees
- C. 270 degrees
- D. 360 degrees
Answer: B. 180 degrees — A linear pair consists of two adjacent angles whose non-common arms form a straight line, meaning their sum is always 180 degrees.
MCQ 9. An angle measures 120 degrees. What is the measure of its supplement?
- A. 30 degrees
- B. 60 degrees
- C. 90 degrees
- D. 180 degrees
Answer: B. 60 degrees — Supplementary angles sum to 180 degrees. So, 180 - 120 = 60 degrees.
MCQ 10. Which of the following statements is true about vertically opposite angles?
- A. They are always complementary.
- B. They always sum to 180 degrees.
- C. They are always equal.
- D. They are always adjacent angles.
Answer: C. They are always equal. — When two lines intersect, the angles opposite each other at the point of intersection are called vertically opposite angles, and they are always equal in measure.
MCQ 11. Two parallel lines are intersected by a transversal. If one interior angle on the same side of the transversal is 70 degrees, what is the measure of the other interior angle on the same side?
- A. 70 degrees
- B. 110 degrees
- C. 90 degrees
- D. 180 degrees
Answer: B. 110 degrees — When a transversal intersects two parallel lines, interior angles on the same side of the transversal are supplementary, meaning their sum is 180 degrees. So, 180 - 70 = 110 degrees.
MCQ 12. If angle A and angle B are complementary, and angle A = 2 * angle B, what is the measure of angle A?
- A. 30 degrees
- B. 45 degrees
- C. 60 degrees
- D. 90 degrees
Answer: C. 60 degrees — Let angle B be x. Then angle A is 2x. Since they are complementary, x + 2x = 90 degrees, which means 3x = 90 degrees, so x = 30 degrees. Therefore, angle A = 2 * 30 = 60 degrees.
MCQ 13. Which of these is NOT a type of angle formed when a transversal intersects two lines?
- A. Corresponding angles
- B. Alternate interior angles
- C. Reflex angles
- D. Vertically opposite angles
Answer: C. Reflex angles — Reflex angles are a type of angle based on their measure (greater than 180 degrees but less than 360 degrees), not a specific pair relationship formed by a transversal with two other lines.
MCQ 14. Lines 'l' and 'm' are parallel, and 't' is a transversal. If a corresponding angle is 105 degrees, what is the measure of its alternate interior angle?
- A. 75 degrees
- B. 105 degrees
- C. 90 degrees
- D. 180 degrees
Answer: B. 105 degrees — When lines are parallel, corresponding angles are equal. Also, alternate interior angles are equal. Therefore, the corresponding angle and the alternate interior angle will both be 105 degrees.
MCQ 15. If two adjacent angles are supplementary, what can be said about their non-common arms?
- A. They are perpendicular.
- B. They are parallel.
- C. They form a straight line.
- D. They are equal in length.
Answer: C. They form a straight line. — When two adjacent angles are supplementary, they form a linear pair, which means their non-common arms extend in opposite directions, forming a straight line.
MCQ 16. Find the value of 'x' if the angles (2x + 10) degrees and (3x - 20) degrees are complementary.
- A. 10 degrees
- B. 20 degrees
- C. 30 degrees
- D. 40 degrees
Answer: C. 30 degrees — For complementary angles, their sum is 90 degrees. So, (2x + 10) + (3x - 20) = 90. This simplifies to 5x - 10 = 90, so 5x = 100, which means x = 20 degrees.
MCQ 17. Two parallel lines are cut by a transversal. If one of the alternate exterior angles is 130 degrees, what is the measure of the interior angle on the same side of the transversal as the given angle?
- A. 50 degrees
- B. 130 degrees
- C. 90 degrees
- D. 180 degrees
Answer: A. 50 degrees — If an alternate exterior angle is 130 degrees, its vertically opposite angle (an interior angle) is also 130 degrees. The interior angle on the same side of the transversal as this 130-degree interior angle would be supplementary to it. So, 180 - 130 = 50 degrees.
MCQ 18. Angles x, y, and z form a straight angle. If angle x = 40 degrees and angle y = 60 degrees, what is the measure of angle z?
- A. 40 degrees
- B. 60 degrees
- C. 80 degrees
- D. 100 degrees
Answer: C. 80 degrees — Angles that form a straight angle sum to 180 degrees. So, x + y + z = 180. Given x = 40 degrees and y = 60 degrees, we have 40 + 60 + z = 180, which means 100 + z = 180. Therefore, z = 180 - 100 = 80 degrees.
MCQ 19. Consider two lines 'p' and 'q' intersected by a transversal 'r'. If a pair of corresponding angles are equal, what can you conclude about lines 'p' and 'q'?
- A. They are perpendicular.
- B. They intersect at two points.
- C. They are parallel.
- D. They are skew lines.
Answer: C. They are parallel. — One of the converse properties of parallel lines is that if a transversal intersects two lines such that a pair of corresponding angles are equal, then the two lines are parallel.
MCQ 20. The measures of two angles are in the ratio 2:7 and they are supplementary. What is the measure of the smaller angle?
- A. 20 degrees
- B. 40 degrees
- C. 70 degrees
- D. 140 degrees
Answer: B. 40 degrees — Let the angles be 2x and 7x. Since they are supplementary, 2x + 7x = 180. This gives 9x = 180, so x = 20 degrees. The smaller angle is 2x = 2 * 20 = 40 degrees.
How to Use These Solutions
Attempt each question yourself first and only then compare with the worked answer. Marks in the CBSE board exam are awarded for the METHOD as much as the final result, so reproduce the steps rather than memorising the last line. Where a solution states a law, a formula or a definition, learn that wording — examiners look for it.
Related practice for this chapter: chapter MCQs, previous-year questions, revision notes and sample papers.