4. Expressions using Letter-Numbers
Answers to all Figure it Out and in-text questions on letter-numbers, formulas, simplifying expressions and pattern rules. CBSE / NCERT Class 7 Ganita Prakash.
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4.1 The Notion of Letter-Numbers
Shabnam’s age = a + 3 = 18 + 3 = 21 years.
Aftab’s age = s − 3 = 20 − 3 = 17 years.
8 coconuts and 9 kg jaggery: 8 × 35 + 9 × 60 = 280 + 540 = ₹820.
Expression: c × 35 + j × 60.
7 coconuts and 4 kg jaggery: 7 × 35 + 4 × 60 = 245 + 240 = ₹485.
4 × q = 4 × 7 = 28 cm.
1. Write formulas for the perimeter.
Let each side be a units.
(a) Equilateral triangle: 3 × a
(b) Regular pentagon: 5 × a
(c) Regular hexagon: 6 × a
2. Combined length of the pipe
20 + k metres
3. Complete the table.
| ₹100 notes | ₹20 notes | ₹5 notes | Expression and total amount |
|---|---|---|---|
| 3 | 5 | 6 | 3 × 100 + 5 × 20 + 6 × 5 = ₹430 |
| 6 | 4 | 3 | 6 × 100 + 4 × 20 + 3 × 5 = ₹695 |
| 8 | 4 | z | 8 × 100 + 4 × 20 + z × 5 = 880 + 5 × z |
| x | y | z | x × 100 + y × 20 + z × 5 |
4. Which expression gives the time to grind y kg of grain?
(d) 10 + 8 × y. 10 seconds to start, then 8 seconds for each kg.
5. Write algebraic expressions (let the number be n).
(a) n + 5
(b) n − 4
(c) 13 × n − 2
(d) 2 × n − 13
6. Describe situations.
(a) 8 × x + 3 × y: A pen costs ₹x and a notebook costs ₹y. Total cost of 8 pens and 3 notebooks.
(b) 15 × j − 2 × k: A box has 15 mangoes. Mohan buys j boxes and gives away k pairs of mangoes. Mangoes left with him.
7. Dates in the 2 × 3 grid when the bottom middle date is w.
Top row: w − 8, w − 7, w − 6
Bottom row: w − 1, w, w + 1
(The date just above is 7 less, because a week has 7 days.)
4.2 Revisiting Arithmetic Expressions
3. 34 − 14 + 20 = 20 + 20 = 40
4. 42 + 15 − (8 − 7) = 57 − 1 = 56
6. 7 × 4 + 9 × 6 = 28 + 54 = 82
7. 20 + 8 × (16 − 6) = 20 + 8 × 10 = 100
4.3 Omission of the Multiplication Symbol
| No. | Given | Mistake? | Correct value |
|---|---|---|---|
| 1 | a = −4, 10 − a = 6 | Yes. They did 10 − 4. | 10 − (−4) = 14 |
| 2 | d = 6, 3d = 36 | Yes. 3d is 3 × d, not 36. | 3 × 6 = 18 |
| 3 | s = 7, 3s − 2 = 15 | Yes. Wrong calculation. | 21 − 2 = 19 |
| 4 | r = 8, 2r + 1 = 29 | Yes. 2r was read as 28. | 16 + 1 = 17 |
| 5 | j = 5, 2j = 10 | No mistake. | 10 |
| 6 | m = −6, 3(m + 1) = 19 | Yes. Sign and bracket mistake. | 3 × (−5) = −15 |
| 7 | f = 3, g = 1, 2f − 2g = 2 | Yes. Wrong calculation. | 6 − 2 = 4 |
| 8 | t = 4, b = 3, 2t + b = 24 | Yes. Multiplied 2 × 4 × 3. | 8 + 3 = 11 |
| 9 | h = 5, n = 6, h − (3 − n) = 4 | Yes. Bracket sign mistake. | 5 − (−3) = 8 |
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4.4 Simplification of Algebraic Expressions
Money from pencils on Day 2 = 3c and on Day 3 = 10c.
If c = ₹50, money from pencils = 18c = 18 × 50 = ₹900.
Money from erasers = 4d + 6d + 1d = 11d.
Check: if c = 2, 5c + 3c + 10c = 10 + 6 + 20 = 36 and 18c = 36. Both are equal.
Yes. Removing brackets and adding like terms gives 34x + 65y.
Yes, we can write it so. Subtracting a bracket is the same as adding its inverse.
Each expression means: (number of correct answers) × p minus (number of wrong answers) × q.
Round 2: 8 × 4 − 4 × 1 = 28. Round 3: 6 × 4 − 2 × 1 = 22.
If there is no penalty, q = 0.
Krishita’s possible rounds: 8p − 2q, 7p − 3q, 8p − 2q (they add up to 23p − 7q).
Krishita scored more. She has more correct answers (23 > 21) and fewer penalties (7 < 9).
Difference: 23p − 7q − (21p − 9q) = 23p − 7q − 21p + 9q = 2p + 2q.
| u | 5u | 5 + u |
|---|---|---|
| 2 | 10 | 7 |
| 5 | 25 | 10 |
| 8 | 40 | 13 |
| 11 | 55 | 16 |
| y | 10y − 3 | 10(y − 3) |
|---|---|---|
| 2 | 17 | −10 |
| 0 | −3 | −30 |
| 7 | 67 | 40 |
| 10 | 97 | 70 |
The values are different, so 5u ≠ 5 + u and 10y − 3 ≠ 10(y − 3).
1. Add the numbers in each picture.
Picture 1: (5y + 5y) + (x + x) + (−6 + 2) = 10y + 2x − 4. Row-wise: (5y − 6 + x) + (x + 2 + 5y) gives the same.
Picture 2: 2p four times, 3q four times, and 3 + 3 − 2 − 2 = 2. Total = 8p + 12q + 2.
Picture 3: −5g four times and 5k twelve times. Total = 60k − 20g.
2. Simplify.
(a) p + p + p + p = 4p; p + p + p + q = 3p + q
(b) p + q + p − q = 2p
(c) p − q + p − q = 2p − 2q
(d) p + q − p + q = 2q
(e) p + q − (p + q) = 0
(f) p − q − p − q = −2q
(g) 2d − d − d − d = −d
(h) 2d − d − d − c = −c
(i) 2d − d − (d − c) = 2d − d − d + c = c
(j) 2d − (d − d) − c = 2d − c
(k) 2d − d − c − c = d − 2c
| No. | Expression | Given | Correct simplest form |
|---|---|---|---|
| 1 | 3a + 2b | 5 | 3a + 2b (unlike terms cannot be added) |
| 2 | 3b − 2b − b | 0 | 0 (correct) |
| 3 | 6(p + 2) | 6p + 8 | 6p + 12 |
| 4 | (4x + 3y) − (3x + 4y) | x + y | x − y |
| 5 | 5 − (2 − 6z) | 3 − 6z | 3 + 6z |
| 6 | 2 + (x + 3) | 2x − 6 | x + 5 |
| 7 | 2y + (3y − 6) | −y + 6 | 5y − 6 |
| 8 | 7p − p + 5q − 2q | 7p + 3q | 6p + 3q |
| 9 | 5(2w + 3x + 4w) | 10w + 15x + 20w | 30w + 15x |
| 10 | 3j + 6k + 9h + 12 | 3(j + 2k + 3h + 4) | 3j + 6k + 9h + 12 (no brackets in the simplest form) |
| 11 | 4(2r + 3s + 5) − 20 | −8r − 12s | 8r + 12s |
Relation: the number of terms is at most one more than the number of letter-numbers.
Each letter-number gives one term, and all plain numbers add up to one term.
4.5 Pick Patterns and Reveal Relationships
Machine 1 (2a − b): 6 and 4 → 2 × 6 − 4 = 8.
Machine 2: the formula is a + b − 2. 5 + 2 − 2 = 5; 8 + 1 − 2 = 7; 9 + 11 − 2 = 18; 10 + 10 − 2 = 18; a + b − 2.
Machine 3: the formula is a × b + 1, that is, ab + 1. 4 × 1 + 1 = 5; 6 × 0 + 1 = 1; 3 × 2 + 1 = 7; 10 × 3 + 1 = 31; ab + 1.
99 ÷ 3 leaves remainder 0 → Design C.
122 ÷ 3 leaves remainder 2 → Design B.
148 ÷ 3 leaves remainder 1 → Design A.
Let the centre number be a. Left is a − 1, right is a + 1, above is a − 7, below is a + 7.
Sum = (a − 7) + (a − 1) + a + (a + 1) + (a + 7) = 5a. So, it is always 5 times the centre number.
Three numbers in a row: (a − 1) + a + (a + 1) = 3a.
A 3 × 3 square with centre a: the sum is 9a.
An X shape: (a − 8) + (a − 6) + a + (a + 6) + (a + 8) = 5a.
Step y has 2y + 1 matchsticks.
Step 33: 2 × 33 + 1 = 67; Step 84: 169; Step 108: 217.
Step 3: 3 horizontal and 4 diagonal. Step 4: 4 horizontal and 5 diagonal.
At Step y: horizontal = y, diagonal = y + 1.
Total = y + (y + 1) = 2y + 1. Yes, they add up to 2y + 1.
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1. Which expression gives the total amount earned?
(a) 30x + 20y
2. How many flags did she give away?
(a) p + q + r. Each customer gets one flag, whether they bought one or both flowers.
3. Snail in the well
(a) In one day and night, it moves u − d cm up. After 10 days and nights: 10 × (u − d) cm, that is, 10u − 10d.
(b) If d > u, the snail slips more than it climbs. It goes down each day and ends up below its starting position.
4. How many km would Radha have cycled after 3 weeks?
Week 1: 7 × 5. Week 2: 7 × (5 + z). Week 3: 7 × (5 + 2z).
Total = 35 + 35 + 7z + 35 + 14z = 105 + 21z km.
5. Fill in the missing blanks on the paths.
w + 2 → −5 → w − 3 → ×3 → 3w − 9
w + 2 → −8 → w − 6 → −4 → w − 10
w + 2 → −4 → w − 2 → ×3 → 3w − 6
6. Train from Yahapur to Vahapur
There are 4 parts of the journey (each t minutes) and 3 stops (2 minutes each).
(a) t = 4: 4 × 4 + 3 × 2 = 22 minutes.
(b) 4t + 6 minutes.
7. Simplify.
(a) 4a + 5b + 10
(b) 9a − 9b − 8a − 4b − 16 = a − 13b − 16
(c) 4x − 6 + 8x + 12 = 12x + 6
(d) 8x − 2x + 3 + 12 = 6x + 15
(e) 8h − 5 − 7h + 9 = h + 4
(f) 23 + 24m − 12n − 8n − 3m − 18 = 21m − 20n + 5
8. Add.
(a) 13d + c − 2
(b) 7f + 4s − 4
(c) 8d − 14c + 9 + 16c − 11 − 9d = 2c − d − 2
(d) −7f − 4s + 3
(e) 0
(f) 0
9. Subtract.
(a) (6a + 9b − 18) − (9a − 6b + 14) = −3a + 15b − 32
(b) (7y − 10 + 3x) − (−15x + 13 − 9y) = 18x + 16y − 23
(c) (11 − 10g + 3h) − (17g + 9 − 7h) = −27g + 10h + 2
(d) (6a − 9b − 18) − (9a − 6b + 14) = −3a − 3b − 32
(e) (−3y + 8 − 3x) − (10x + 2 + 10y) = −13x − 13y + 6
(f) (7h − 8g + 20) − (8g + 4h − 10) = −16g + 3h + 30
10. Describe situations.
(a) 8x + 3y: A mango costs ₹8 and a banana costs ₹3. Cost of x mangoes and y bananas.
(b) 15x − 2x: Raju earns ₹15 a day and spends ₹2 a day. Money he saves in x days.
11. Rope folded and cut
No fold → 2 pieces, 1 fold → 3 pieces, 2 folds → 4 pieces.
Folded 10 times → 12 pieces. Folded r times → r + 2 pieces.
12. Matchstick squares
The first square needs 4 sticks. Each new square needs 3 more.
10 squares: 4 + 3 × 9 = 31. w squares: 4 + 3 × (w − 1) = 3w + 1.
13. Traffic signal colours (red, yellow, green, yellow, …)
Red at 1, 5, 9, … → 4n − 3. Green at 3, 7, 11, … → 4n − 1. Yellow at 2, 4, 6, … → 2n.
Position 90: even → yellow. Position 190: even → yellow. Position 343 = 4 × 86 − 1 → green.
14. Squares in the pattern
Step 1: 5, Step 2: 9, Step 3: 13. Each step adds 4 squares.
Step 4: 17, Step 10: 41, Step 50: 201. Formula: 4n + 1.
Vertices: each square has 4 vertices, so 4 × (4n + 1) = 16n + 4. (If shared corners are counted once, it is 12n + 4.)
15. Endless 4-column grid
(a) Column 1: 4r − 3; Column 2: 4r − 2; Column 3: 4r − 1; Column 4: 4r (r = row number).
(b) (i) 124 = 4 × 31 → row 31, column 4. (ii) 147 = 4 × 37 − 1 → row 37, column 3. (iii) 201 = 4 × 51 − 3 → row 51, column 1.
(c) The number in row r and column c is 4(r − 1) + c, that is, 4r − 4 + c.
(d) Multiples of 3 appear in columns 3, 2, 1, 4, 3, 2, 1, 4, … in turn.
Other patterns: column 4 has multiples of 4; odd numbers are in columns 1 and 3; even numbers are in columns 2 and 4.