Chapter 4 exercise answers: Expressions using Letter-Numbers

Open NCERT p. 84Checked by computerAnswers can differ: one example

(a) triangle with all sides equal

1. Call the side length s.
2. A triangle with all sides equal has 3 sides, each s.
3. Perimeter = s + s + s = 3s.

Answer3s

(b) a regular pentagon

1. A regular pentagon has 5 equal sides, each s.
2. Perimeter = 5 × s.

Answer5s

(c) a regular hexagon

1. A regular hexagon has 6 equal sides, each s.
2. Perimeter = 6 × s.

Answer6s

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1. Munirathna already has a pipe 20 m long.
2. He joins another pipe of length k meters to it.
3. Joining pipes end to end adds their lengths.
4. Combined length = 20 + k.

Answer(20 + k) meters

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1. Multiply the count of each note by its value, and add.
2. Row 1: 3×100 + 5×20 + 6×5 = 300+100+30 = ₹430.
3. Row 2's total is already given as 6×100 + 4×20 + 3×5 = ₹695, so it has 6 ₹100-notes, 4 ₹20-notes and 3 ₹5-notes.
4. Row 3 has 8 ₹100-notes, 4 ₹20-notes and z ₹5-notes: 8×100 + 4×20 + z×5 = 880 + 5z.
5. Row 4 has x ₹100-notes, y ₹20-notes and z ₹5-notes: 100x + 20y + 5z.

AnswerRow 1: ₹430. Row 2: 6, 4 and 3 notes. Row 3: 880 + 5z. Row 4: 100x + 20y + 5z.

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1. Starting the machine always takes 10 seconds, once only.
2. Grinding y kg then takes 8 seconds per kg: 8 × y.
3. Total time = start-up time + grinding time = 10 + 8y.
4. This matches option (d).

Answer(d) 10 + 8 × y

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(a) 5 more than a number

1. Call the number n.
2. 5 more than n means add 5.

Answern + 5

(b) 4 less than a number

1. Call the number n.
2. 4 less than n means subtract 4.

Answern − 4

(c) 2 less than 13 times a number

1. Call the number n.
2. 13 times the number is 13n.
3. 2 less than that is 13n − 2.

Answer13n − 2

(d) 13 less than 2 times a number

1. Call the number n.
2. 2 times the number is 2n.
3. 13 less than that is 2n − 13.

Answer2n − 13

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Open NCERT p. 85One way to think about it

(a) 8 × x + 3 × y

1. Many different situations fit this expression; here is one.
2. Let x be the number of pencils bought at ₹8 each, so 8 × x is the cost of the pencils in rupees.
3. Let y be the number of erasers bought at ₹3 each, so 3 × y is the cost of the erasers.
4. 8 × x + 3 × y is the total cost. For example, 3 pencils and 4 erasers cost 8 × 3 + 3 × 4 = 24 + 12 = ₹36.

In shortThe total cost, in rupees, of x pencils at ₹8 each and y erasers at ₹3 each.

(b) 15 × j – 2 × k

1. Many different situations fit this expression; here is one.
2. In a game you win 15 points for every jump you land and lose 2 points for every fall. Let j be the number of jumps landed and k the number of falls.
3. 15 × j is the points won and 2 × k is the points lost.
4. 15 × j − 2 × k is the final score. For example, 4 jumps and 5 falls give 15 × 4 − 2 × 5 = 60 − 10 = 50 points.

In shortThe final score of a player who wins 15 points for each of j jumps and loses 2 points for each of k falls.

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1. The bottom row runs one date at a time: bottom-left = w−1, bottom-middle = w (given), bottom-right = w+1.
2. The top row sits exactly one week (7 days) earlier than the same column of the bottom row.
3. Top-left = (w−1) − 7 = w−8.
4. Top-middle = w − 7.
5. Top-right = (w+1) − 7 = w−6.

AnswerTop-left = w − 8, top-middle = w − 7, top-right = w − 6, bottom-right = w + 1.

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First picture

1. The numbers are 5y, −6, x in the top row and x, 2, 5y in the bottom row.
2. Row by row: (5y − 6 + x) + (x + 2 + 5y).
3. Put like terms together: x + x = 2x, 5y + 5y = 10y, and −6 + 2 = −4.

Answer2x + 10y − 4

Second picture

1. The numbers are 2p, 3q, −2, 3 (top row), 3q, 2p, 3, −2 (second row), then 2p, 3q and 3q, 2p.
2. There are four 2p's: 2p + 2p + 2p + 2p = 8p.
3. There are four 3q's: 3q + 3q + 3q + 3q = 12q.
4. The plain numbers: −2 + 3 + 3 − 2 = 2.

Answer8p + 12q + 2

Third picture

1. The top and bottom rows are each −5g, 5k, 5k, −5g; the two middle rows are each 5k, 5k, 5k, 5k.
2. There are four −5g's: 4 × (−5g) = −20g.
3. There are twelve 5k's: 12 × 5k = 60k.
4. Another way: the top two rows add to −10g + 30k and the bottom two rows are the same, so the sum is 2 × (−10g + 30k) = −20g + 60k.

Answer−20g + 60k

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(a) p + p + p + p

1. p+p+p+p = 4p (four p's added).
2. p+p+p+q = 3p+q (three p's, one q).

Answer4p; 3p + q

(b) p + q + p – q

1. +q and −q cancel out.
2. Left with p+p = 2p.

Answer2p

(c) p – q + p – q

1. Two p's add to 2p.
2. Two −q's add to −2q.

Answer2p − 2q

(d) p + q – p + q

1. p and −p cancel out.
2. Two q's remain: q+q = 2q.

Answer2q

(e) p + q – (p + q)

1. Remove the bracket: p+q−p−q.
2. p and −p cancel; q and −q cancel.

Answer0

(f) p – q – p – q

1. p and −p cancel out.
2. −q−q = −2q remains.

Answer−2q

(g) 2d – d – d – d

1. 2d−d = d.
2. d−d = 0.
3. 0−d = −d.

Answer−d

(h) 2d – d – d – c

1. 2d−d−d = 0.
2. 0−c = −c.

Answer−c

(i) 2d – d – (d – c)

1. Remove the bracket: the minus flips −c inside to +c, giving 2d−d−d+c.
2. 2d−d−d = 0.
3. 0+c = c.

Answerc

(j) 2d – (d – d) – c

1. d−d inside the bracket is 0.
2. 2d−0−c = 2d−c.

Answer2d − c

(k) 2d – d – c – c

1. 2d−d = d.
2. −c−c = −2c.

Answerd − 2c

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1. Jowar roti earns ₹30 per plate, for x plates: 30x.
2. Pulao earns ₹20 per plate, for y plates: 20y.
3. Total = 30x + 20y, which is option (a).
4. The other options swap the prices, multiply the wrong quantities together, or subtract instead of add, so none of them match.

Answer(a) 30x + 20y

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1. Every customer gets exactly one flag, no matter what they bought.
2. Total customers = only-champak (p) + only-marigold (q) + both (r).
3. So flags given away = p + q + r, which is option (a).

Answer(a) p + q + r

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(a) how far away the snail is from its starting position

1. Measure the snail's height from where it started, counting up as positive.
2. Each day it climbs u cm and each night it slips d cm, so one day and one night change its height by u − d.
3. Over 10 days and 10 nights: 10 × (u − d) = 10u − 10d.

Answer10u − 10d cm above its starting position (a negative value means it is below the start)

(b) What can we say about the snail's movement if d > u?

1. If d > u, the snail slips more each night than it climbs each day, so u − d is negative.
2. Then 10u − 10d is negative too. For example, u = 3 and d = 5 give 30 − 50 = −20, which is 20 cm below the start.
3. So overall the snail moves down.

AnswerThe snail moves down: after 10 days and nights it is 10d − 10u cm below where it started.

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1. Week 1: 5 km a day for 7 days = 35 km.
2. Week 2: daily distance is (5+z) km, for 7 days = 35 + 7z km.
3. Week 3: daily distance is (5+2z) km, for 7 days = 35 + 14z km.
4. Add the three weeks: 35 + (35+7z) + (35+14z) = 105 + 21z.

Answer105 + 21z km

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1. Start from the centre, w + 2, and apply each box's operation in order, moving outward.
2. Top-left: (w+2) − 5 = w − 3 (given), then × 3 gives 3w − 9.
3. Bottom-left: (w+2) − 8 = w − 6, then − 4 gives w − 10.
4. Bottom-right: (w+2) − 4 = w − 2.

AnswerTop-left end oval: 3w − 9. Bottom-left middle oval: w − 6. Bottom-left end oval: w − 10. Bottom-right middle oval: w − 2.

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(a) If t = 4, what is the time taken …

1. Three stations along the way split the trip into 4 equal legs.
2. Each leg takes t minutes, so travelling time = 4t.
3. The train also stops 2 minutes at each of the 3 stations: 3×2 = 6 minutes.
4. Total = 4t + 6; with t = 4, that's 16 + 6 = 22 minutes.

Answer22 minutes

(b) What is the algebraic expression for the time taken …

1. Same reasoning in general: 4 legs of t minutes give 4t.
2. Add the fixed 6 minutes of stops.

Answer4t + 6

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(a) 3a + 9b – 6 + 8a – 4b – 7a …

1. a-terms: 3a+8a−7a = 4a.
2. b-terms: 9b−4b = 5b.
3. Numbers: −6+16 = 10.

Answer4a + 5b + 10

(b) 3 (3a – 3b) – 8a – 4b – 16

1. Multiply out: 3(3a−3b) = 9a−9b.
2. a-terms: 9a−8a = a.
3. b-terms: −9b−4b = −13b.
4. Numbers: −16.

Answera − 13b − 16

(c) 2 (2x – 3) + 8x + 12

1. Multiply out: 2(2x−3) = 4x−6.
2. x-terms: 4x+8x = 12x.
3. Numbers: −6+12 = 6.

Answer12x + 6

(d) 8x – (2x – 3) + 12

1. Remove the bracket: −(2x−3) = −2x+3.
2. x-terms: 8x−2x = 6x.
3. Numbers: 3+12 = 15.

Answer6x + 15

(e) 8h – (5 + 7h) + 9

1. Remove the bracket: −(5+7h) = −5−7h.
2. h-terms: 8h−7h = h.
3. Numbers: −5+9 = 4.

Answerh + 4

(f) 23 + 4(6m – 3n) – 8n – 3m – 18

1. Multiply out: 4(6m−3n) = 24m−12n.
2. m-terms: 24m−3m = 21m.
3. n-terms: −12n−8n = −20n.
4. Numbers: 23−18 = 5.

Answer21m − 20n + 5

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(a) 4d – 7c + 9 and 8c – 11 + 9d

1. d-terms: 4d+9d = 13d.
2. c-terms: −7c+8c = c.
3. Numbers: 9−11 = −2.

Answer13d + c − 2

(b) – 6f + 19 – 8s and – 23 + 13f …

1. f-terms: −6f+13f = 7f.
2. s-terms: −8s+12s = 4s.
3. Numbers: 19−23 = −4.

Answer7f + 4s − 4

(c) 8d – 14c + 9 and 16c – (11 + 9d)

1. Remove the bracket in the second expression: 16c−11−9d.
2. d-terms: 8d−9d = −d.
3. c-terms: −14c+16c = 2c.
4. Numbers: 9−11 = −2.

Answer−d + 2c − 2

(d) 6f – 20 + 8s and 23 – 13f – 12s

1. f-terms: 6f−13f = −7f.
2. s-terms: 8s−12s = −4s.
3. Numbers: −20+23 = 3.

Answer−7f − 4s + 3

(e) 13m – 12n and 12n – 13m

1. m-terms: 13m−13m = 0.
2. n-terms: −12n+12n = 0.

Answer0

(f) – 26m + 24n and 26m – 24n

1. m-terms: −26m+26m = 0.
2. n-terms: 24n−24n = 0.

Answer0

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(a) 9a – 6b + 14 from 6a + 9b – 18

1. Flip every sign of (9a−6b+14): −9a+6b−14.
2. Add to 6a+9b−18: a-terms 6a−9a = −3a.
3. b-terms 9b+6b = 15b.
4. Numbers −18−14 = −32.

Answer−3a + 15b − 32

(b) – 15x + 13 – 9y from 7y – 10 + 3x

1. Flip every sign of (−15x+13−9y): 15x−13+9y.
2. Add to 7y−10+3x: x-terms 3x+15x = 18x.
3. y-terms 7y+9y = 16y.
4. Numbers −10−13 = −23.

Answer18x + 16y − 23

(c) 17g + 9 – 7h from 11 – 10g + 3h

1. Flip every sign of (17g+9−7h): −17g−9+7h.
2. Add to 11−10g+3h: g-terms −10g−17g = −27g.
3. h-terms 3h+7h = 10h.
4. Numbers 11−9 = 2.

Answer−27g + 10h + 2

(d) 9a – 6b + 14 from 6a – (9b + 18)

1. Remove the bracket in 6a−(9b+18): 6a−9b−18.
2. Flip every sign of (9a−6b+14): −9a+6b−14.
3. Add: a-terms 6a−9a = −3a; b-terms −9b+6b = −3b; numbers −18−14 = −32.

Answer−3a − 3b − 32

(e) 10x + 2 + 10y from –3y + 8 – 3x

1. Flip every sign of (10x+2+10y): −10x−2−10y.
2. Add to −3y+8−3x: x-terms −3x−10x = −13x.
3. y-terms −3y−10y = −13y.
4. Numbers 8−2 = 6.

Answer−13x − 13y + 6

(f) 8g + 4h – 10 from 7h – 8g + 20

1. Flip every sign of (8g+4h−10): −8g−4h+10.
2. Add to 7h−8g+20: g-terms −8g−8g = −16g.
3. h-terms 7h−4h = 3h.
4. Numbers 20+10 = 30.

Answer−16g + 3h + 30

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Open NCERT p. 103One way to think about it

(a) 8x + 3y

1. Many different situations fit this expression; here is one.
2. Meena walks 8 km a day for x days, which is 8x km.
3. She cycles 3 km a day for y days, which is 3y km.
4. 8x + 3y is the total distance she covers. For example, x = 2 and y = 5 give 16 + 15 = 31 km.

In shortThe total distance, in km, covered by walking 8 km a day for x days and cycling 3 km a day for y days.

(b) 15x – 2x

1. Many different situations fit this expression; here is one. Both terms use the same letter x, so the situation must use the same number twice.
2. x children each pay ₹15 for a school trip, so ₹15x is collected.
3. Each child is then given ₹2 back, so ₹2x is returned.
4. 15x − 2x is the money the school keeps. It simplifies to 13x, since each child really pays ₹13. For example, 10 children: 150 − 20 = 130 = 13 × 10.

In shortThe money kept when x children each pay ₹15 and each get ₹2 back; this equals 13x rupees.

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1. Cut a straight rope once: 2 pieces (first picture).
2. Fold it once into a U: the cut goes through 2 strands and gives 3 pieces (second picture).
3. Fold it twice, back and forth as in the third picture: the cut goes through 3 strands and gives 4 pieces.
4. In general, r folds made this way lay r + 1 strands side by side. The one cut goes through every strand, so the rope is cut in r + 1 places, and r + 1 cuts split a rope into r + 2 pieces.
5. For 10 folds: 10 + 2 = 12 pieces.

Answer12 pieces for 10 folds; r + 2 pieces for r folds.

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1. The first square alone takes 4 matchsticks.
2. Each square after the first shares one side with the square before it, so it only needs 3 more matchsticks.
3. For w squares: 4 for the first, plus 3 for each of the remaining (w−1): 4 + 3(w−1) = 3w + 1.
4. For w = 10: 3×10 + 1 = 31.

Answer31 matchsticks for 10 squares; in general, 3w + 1.

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1. The sequence red, yellow, green, yellow repeats every 4 positions.
2. Every EVEN position is yellow (it sits at both position 2 and position 4 of each block of 4).
3. The odd positions split between red (1, 5, 9, … i.e. 4n−3) and green (3, 7, 11, … i.e. 4n−1).
4. 90 is even → yellow. 190 is even → yellow. 343 is odd, and 343 = 4×86 − 1 → green.

AnswerPosition 90: yellow. Position 190: yellow. Position 343: green. Red: 4n−3. Yellow: 2n. Green: 4n−1 (n = 1, 2, 3, …).

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1. Count the squares: Step 1 has 5, Step 2 has 9, Step 3 has 13. Each step adds 4 squares, one at the end of each of the 4 arms.
2. In Step n there is 1 square in the middle and n squares on each of the 4 arms, so the number of squares is 4n + 1.
3. Step 4: 4 × 4 + 1 = 17. Step 10: 4 × 10 + 1 = 41. Step 50: 4 × 50 + 1 = 201.
4. Vertices: every square has 4 vertices. In the picture the squares are drawn with a small gap between them, so no two squares share a vertex. The number of vertices is 4 × (4n + 1) = 16n + 4.
5. If instead you take neighbouring squares to touch at their corners, the squares meet at 4n points (n on each arm), and each of these points is a vertex of two squares. Then the count is 16n + 4 − 4n = 12n + 4.

AnswerStep 4: 17 squares, Step 10: 41 squares, Step 50: 201 squares. Formula: 4n + 1. Vertices, with the squares separate as drawn: 4 × (4n + 1) = 16n + 4. Vertices, if neighbouring squares share a corner: 12n + 4.

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Open NCERT p. 105Matches NCERT’s answer

(a) Give expressions to generate all the numbers in a given column

1. Row r starts right after 4(r−1) numbers are already written.
2. Column 1: 4(r−1)+1 = 4r−3.
3. Column 2: 4r−2. Column 3: 4r−1. Column 4: 4r.

AnswerColumn 1: 4r−3; column 2: 4r−2; column 3: 4r−1; column 4: 4r.

(b)(i) 124

1. 124 ÷ 4 = 31 exactly, so it's the last (column 4) number of row 31.

AnswerRow 31, column 4

(b)(ii) 147

1. 147 = 4×37 − 1, matching the column-3 formula with r = 37.

AnswerRow 37, column 3

(b)(iii) 201

1. 201 = 4×51 − 3, matching the column-1 formula with r = 51.

AnswerRow 51, column 1

(c) What number appears in row r and column c?

1. Row r begins right after 4(r−1) numbers have already been written.
2. Column c then adds c more: 4(r−1) + c.

Answer4(r − 1) + c

(d) Observe the positions of multiples of 3.

1. Listing the columns of 3, 6, 9, 12, 15, 18, 21, 24, …: 3, 2, 1, 4, 3, 2, 1, 4, …
2. This block of 4 keeps repeating because every 4th multiple of 3 lines up with the 4-column grid again.

Answer3, 2, 1, 4 (then it repeats)

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