Chapter 7 exercise answers: Finding the Unknown

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(a) 3x – 10 = 35

1. 3x − 10 = 35
2. Add 10 to both sides: 3x = 45
3. Divide both sides by 3: x = 15

Answerx = 15

(b) 5s = 3s

1. 5s = 3s
2. Subtract 3s from both sides: 2s = 0
3. Divide both sides by 2: s = 0

Answers = 0

(c) 3u – 7 = 2u + 3

1. 3u − 7 = 2u + 3
2. Subtract 2u from both sides: u − 7 = 3
3. Add 7 to both sides: u = 10

Answeru = 10

(d) 4 (m + 6) – 8 = 2m – 4

1. 4(m + 6) − 8 = 2m − 4
2. Open the bracket: 4m + 24 − 8 = 2m − 4
3. 4m + 16 = 2m − 4
4. Subtract 2m from both sides: 2m + 16 = −4
5. Subtract 16 from both sides: 2m = −20
6. Divide both sides by 2: m = −10

Answerm = −10

(e) u/15 = 6

1. u/15 = 6
2. Multiply both sides by 15: u = 90

Answeru = 90

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Open NCERT p. 172Checked by computerAnswers can differ: one example

1. Use the hint: 4 more than a number and 5 more than the same number can never be equal.
2. Write it as an equation: x + 4 = x + 5.
3. Subtract x from both sides: 4 = 5, which is never true.
4. So no value of x can make the two sides equal.

Answerx + 4 = x + 5 (no solution, since it reduces to 4 = 5)

Open NCERT p. 181Checked by computerAnswers can differ: one example

1. Start from x = −2 and do the same thing to both sides each time.
2. Add 2 to both sides: x + 2 = 0.
3. Double both sides of x = −2 first, then add 4: 2x + 4 = 0.
4. Subtract 5 from both sides: x − 5 = −7.
5. Multiply by 5 then add 3: 5x + 3 = −7.
6. Multiply by −3: −3x = 6.

Answerx + 2 = 0, 2x + 4 = 0, x − 5 = −7, 5x + 3 = −7, −3x = 6

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(a) 2y = 60

1. 2y = 60
2. Divide both sides by 2: y = 30

Answery = 30

(b) –8 = 5x – 3

1. −8 = 5x − 3
2. Add 3 to both sides: −5 = 5x
3. Divide both sides by 5: x = −1

Answerx = −1

(c) –53w = –15

1. −53w = −15
2. Divide both sides by −53: w = 15/53

Answerw = 15/53

(d) 13 – z = 8

1. 13 − z = 8
2. Subtract 13 from both sides: −z = −5
3. So z = 5

Answerz = 5

(e) k + 8 = 12 – k

1. k + 8 = 12 − k
2. Add k to both sides: 2k + 8 = 12
3. Subtract 8: 2k = 4
4. Divide by 2: k = 2

Answerk = 2

(f) 7m = m – 3

1. 7m = m − 3
2. Subtract m from both sides: 6m = −3
3. Divide both sides by 6: m = −1/2

Answerm = −1/2

(g) 3n = 10 + n

1. 3n = 10 + n
2. Subtract n from both sides: 2n = 10
3. Divide both sides by 2: n = 5

Answern = 5

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1. Let the unit's digit be u.
2. The ten's digit is 3 less than the unit's digit: ten's digit = u − 3.
3. The hundred's digit is 3 less than the ten's digit: hundred's digit = u − 6.
4. The three digits add to 15: (u − 6) + (u − 3) + u = 15.
5. 3u − 9 = 15, so 3u = 24, so u = 8.
6. Ten's digit = 8 − 3 = 5. Hundred's digit = 8 − 6 = 2.
7. The number is 258.

Answer258

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1. Let the weight of the brick be w kg.
2. Half its weight plus 1 kg equals the whole weight: w/2 + 1 = w.
3. Subtract w/2 from both sides: 1 = w/2.
4. Multiply both sides by 2: w = 2.

Answer2 kg

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1. Let the number be n.
2. One quarter of it, increased by 9, equals the number itself: n/4 + 9 = n.
3. Subtract n/4 from both sides: 9 = n − n/4 = 3n/4.
4. Multiply both sides by 4/3: n = 12.

Answer12

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(c) k

1. 4k + 1 = 13
2. Subtract 1 from both sides: 4k = 12
3. Divide both sides by 4: k = 3

Answerk = 3

(b) 4k

1. From 4k + 1 = 13, subtracting 1 gives 4k = 12 directly.

Answer4k = 12

(a) 8k + 2

1. 8k + 2 is twice (4k + 1): 8k + 2 = 2 × 13 = 26.

Answer8k + 2 = 26

(d) 4k – 1

1. 4k − 1 = (4k) − 1 = 12 − 1 = 11, using 4k = 12.

Answer4k − 1 = 11

(e) –k – 2

1. k = 3, so −k − 2 = −3 − 2 = −5.

Answer−k − 2 = −5

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(a) 5 × ___ – 8 = 37

1. 5 × ___ − 8 = 37
2. Add 8 to both sides: 5 × ___ = 45
3. Divide by 5: ___ = 9

Answer9

(b) 37 – (33 – ___ ) = 35

1. 37 − (33 − ___) = 35
2. So 33 − ___ = 37 − 35 = 2
3. So ___ = 33 − 2 = 31

Answer31

(c) – 3 × (– 11 + ___ ) = 45

1. −3 × (−11 + ___) = 45
2. So −11 + ___ = 45 ÷ (−3) = −15
3. So ___ = −15 + 11 = −4

Answer−4

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1. Let the number of ₹50 notes (equal to the number of ₹100 notes) be a.
2. Total pay: 50a + 100a = 750.
3. 150a = 750.
4. Divide both sides by 150: a = 5.

Answer5 notes of ₹50 and 5 notes of ₹100

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Open NCERT p. 185Checked by computerAnswers can differ: one example

1. Count the dots that are already showing outside the blobs: there are 4 visible blue dots.
2. Let x be the number of dots hidden under one blob; there are 3 blobs.
3. Total dots: 3x + 4 = 25.
4. Subtract 4 from both sides: 3x = 21.
5. Divide both sides by 3: x = 7.

Answer7 dots under one blob; equation: 3x + 4 = 25

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(a) output 43

1. The worked example shows: input → +3 → ×4 → −5 = output, so output = 4(x + 3) − 5.
2. Set 4(x + 3) − 5 = 43.
3. 4(x + 3) = 48, so x + 3 = 12, so x = 9.

Answer9

(a) output 75

1. 4(x + 3) − 5 = 75.
2. 4(x + 3) = 80, so x + 3 = 20, so x = 17.

Answer17

(b) output 63

1. The worked example shows the output is (input × 3) − (input + 3), which simplifies to 2x − 3.
2. Check: for input 12, 2(12) − 3 = 21, matching the picture.
3. Set 2x − 3 = 63, so 2x = 66, so x = 33.

Answer33

(b) output 227

1. 2x − 3 = 227, so 2x = 230, so x = 115.

Answer115

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÷3, ÷3 machine

1. Dividing by 3 twice means dividing by 9 overall: x/9 = 5.
2. Multiply both sides by 9: x = 45.

Answer45

−4, −4 machine

1. Subtracting 4 twice: x − 4 − 4 = −11, so x − 8 = −11.
2. Add 8 to both sides: x = −3.

Answer−3

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1. Let the distance travelled be k km.
2. Total cost: 800 + 20k = 2200.
3. Subtract 800 from both sides: 20k = 1400.
4. Divide both sides by 20: k = 70.

Answer70 km

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1. Let the smaller number be x, so the larger number is 3x.
2. Their sum is 76: x + 3x = 76.
3. 4x = 76.
4. Divide both sides by 4: x = 19.
5. The larger number is 3 × 19 = 57.

Answer19 and 57

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Open NCERT p. 187Checked by computerReads two ways: both answers shown

1. Read the drawing from top to bottom: a frame strip of 3 cm, then 6 gaps with 5 rods between them, then a strip at the bottom. Each rod is 2 cm. The 34 cm is the whole height, from the top of the frame to the bottom.
2. The bottom strip has no label, but it is drawn about as thick as the top strip, so take it as 3 cm.
3. Let each gap be x cm. Then 3 + 6x + 5 × 2 + 3 = 34.
4. 6x + 16 = 34, so 6x = 18 and x = 3.
5. If you take the bottom strip to be as thick as a rod (2 cm) instead: 3 + 6x + 5 × 2 + 2 = 34, so 6x = 19 and x = 19/6 = 3 1/6.

AnswerWith the bottom strip 3 cm like the top strip (as drawn): the gap is 3 cm. With the bottom strip taken as 2 cm like a rod: 19/6 cm.

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1. Let the milkshake cost ₹m, so the juice costs ₹(m − 15).
2. 4 juices and 7 milkshakes cost ₹600: 4(m − 15) + 7m = 600.
3. 4m − 60 + 7m = 600.
4. 11m − 60 = 600, so 11m = 660.
5. Divide both sides by 11: m = 60.
6. Juice = 60 − 15 = 45.

AnswerFruit juice ₹45, chocolate milkshake ₹60

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1. 28p − 36 = 98, so 28p = 134, so 14p = 67.
2. 14p − 19 = 67 − 19 = 48.
3. 28p − 38 = 134 − 38 = 96.

Answer14p − 19 = 48, 28p − 38 = 96

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(a)

1. Mistake: both sides were divided by 6, but only 6x and 66 were divided — the 9 was left as it was. Dividing the whole left side by 6 would also divide the 9.
2. Correct way: 6x + 9 = 66. Subtract 9 from both sides: 6x = 57.
3. Divide both sides by 6: x = 57/6 = 19/2.
4. Check: 6 × 19/2 + 9 = 57 + 9 = 66.

Answerx = 19/2 (not 2)

(b)

1. Divide both sides by 2: 7y + 12 = 18. Correct.
2. Subtract 12 from both sides: 7y = 6. Correct.
3. Divide both sides by 7: y = 6/7. Correct.
4. There is no mistake in (b).

Answery = 6/7 (already correct)

(c)

1. Mistake 1: to remove −5 from the left side, add 5 to both sides. The right side becomes 9x + 8 + 5, not 9x + 8 − 5.
2. Mistake 2: from −5x = 3, dividing both sides by −5 gives x = 3/(−5) = −3/5, not −5/3. The fraction was turned upside down.
3. Correct way: 4x − 5 = 9x + 8. Add 5 to both sides: 4x = 9x + 13.
4. Subtract 9x from both sides: −5x = 13.
5. Divide both sides by −5: x = −13/5.
6. Check: 4 × (−13/5) − 5 = −77/5 and 9 × (−13/5) + 8 = −77/5.

Answerx = −13/5

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Triangle 1

1. The tick marks show the two slanting sides are equal, so the angles opposite them — the two base angles — are equal. Each is (y + 15).
2. The angles of a triangle add up to 180°: y + (y + 15) + (y + 15) = 180.
3. 3y + 30 = 180, so 3y = 150, so y = 50.
4. Each base angle is 50 + 15 = 65.
5. Check: 50 + 65 + 65 = 180.

Answery = 50°; the base angles are 65° and 65°

Triangle 2

1. The angles of a triangle add up to 180°: x + (x − 10) + (x + 10) = 180.
2. The −10 and +10 cancel: 3x = 180, so x = 60.
3. The other two angles are 60 − 10 = 50 and 60 + 10 = 70.
4. Check: 60 + 50 + 70 = 180.

Answerx = 60°; the other angles are 50° and 70°

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Open NCERT p. 188Checked by computerAnswers can differ: one example

1. Start from u = 6 and do the same thing to both sides each time.
2. Subtract 6 from both sides: u − 6 = 0.
3. Multiply both sides by 2: 2u = 12.
4. Add 4 to both sides: u + 4 = 10.
5. Multiply by 3 then subtract 2: 3u − 2 = 16.

Answeru − 6 = 0, 2u = 12, u + 4 = 10, 3u − 2 = 16

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1. Let the first person get a.
2. The second person gets 2a.
3. The third person gets 3 times the second: 3 × 2a = 6a.
4. The fourth person gets 4 times the third: 4 × 6a = 24a.
5. Total: a + 2a + 6a + 24a = 132.
6. 33a = 132.
7. Divide both sides by 33: a = 4.

Answer4

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1. Let the giraffe's height be h metres.
2. Half its height, plus two and a half metres, equals its full height: h/2 + 5/2 = h.
3. Subtract h/2 from both sides: 5/2 = h/2.
4. Multiply both sides by 2: h = 5.

Answer5 metres

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Figure 1

1. Position n is a row of n squares with a triangle tip on the right, so position n has n squares.
2. Sticks: positions 1, 2, 3 use 6, 9, 12. A row of n squares uses 3n + 1 sticks and the tip adds 2 more, so position n uses 3n + 3 sticks.
3. (a) Position 11 has 11 squares.
4. (b) 3 × 11 + 3 = 36 sticks.
5. (c) 3n + 3 = 85 gives 3n = 82, so n = 82/3. That is not a whole number, so no arrangement uses exactly 85 sticks.
6. (d) 3n + 3 = 150 gives 3n = 147, so n = 49. Yes: position 49.

Answer(a) 11 (b) 36 (c) No (d) Yes, position 49

Figure 2

1. Position n is a staircase: a row of 2 squares at the top, then n − 1 rows of 3 squares, then a row of 2 at the bottom. Positions 1, 2, 3, 4 have 4, 7, 10, 13 squares.
2. Squares: 2 + 3(n − 1) + 2 = 3n + 1.
3. Sticks: the top row of 2 squares needs 7 sticks. Each row of 3 below needs 10 but shares 1 stick with the row above, so it adds 9. The bottom row of 2 needs 7 but shares 1, so it adds 6. Position n uses 7 + 9(n − 1) + 6 = 9n + 4 sticks (13, 22, 31, … in the picture).
4. (a) 3 × 11 + 1 = 34 squares.
5. (b) 9 × 11 + 4 = 103 sticks.
6. (c) 9n + 4 = 85 gives 9n = 81, so n = 9. Yes: position 9.
7. (d) 9n + 4 = 150 gives 9n = 146, so n = 146/9. That is not a whole number, so no arrangement uses exactly 150 sticks.

Answer(a) 34 (b) 103 (c) Yes, position 9 (d) No

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1. Let the number be n.
2. n + 36 = 10n.
3. Subtract n from both sides: 36 = 9n.
4. Divide both sides by 9: n = 4.

Answer4

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(a) 5(r + 2) = 10

1. 5(r + 2) = 10
2. Divide both sides by 5: r + 2 = 2
3. Subtract 2: r = 0

Answerr = 0

(b) – 3(u + 2) = 2(u – 1)

1. −3(u + 2) = 2(u − 1)
2. Open the brackets: −3u − 6 = 2u − 2
3. Add 3u and add 2 to both sides: −4 = 5u
4. Divide by 5: u = −4/5

Answeru = −4/5

(c) 2(7 – 2n) = – 6

1. 2(7 − 2n) = −6
2. Divide both sides by 2: 7 − 2n = −3
3. Subtract 7: −2n = −10
4. Divide by −2: n = 5

Answern = 5

(d) 2(x – 4) = – 16

1. 2(x − 4) = −16
2. Divide both sides by 2: x − 4 = −8
3. Add 4: x = −4

Answerx = −4

(e) 6(x – 1) = 2(x – 1) – 4

1. 6(x − 1) = 2(x − 1) − 4
2. Open the brackets: 6x − 6 = 2x − 2 − 4
3. 6x − 6 = 2x − 6
4. Subtract 2x and add 6 to both sides: 4x = 0
5. So x = 0

Answerx = 0

(f) 3 – 7s = 7 – 3s

1. 3 − 7s = 7 − 3s
2. Add 7s and subtract 7 from both sides: −4 = 4s
3. Divide by 4: s = −1

Answers = −1

(g) 2x + 1 = 6 – (2x – 3)

1. 2x + 1 = 6 − (2x − 3)
2. Open the bracket on the right: 6 − 2x + 3 = 9 − 2x
3. 2x + 1 = 9 − 2x
4. Add 2x to both sides: 4x + 1 = 9
5. Subtract 1: 4x = 8
6. Divide by 4: x = 2

Answerx = 2

(h) 10 – 5x = 3(x – 4) – 2(x – 7)

1. 10 − 5x = 3(x − 4) − 2(x − 7)
2. Open the brackets: 3x − 12 − 2x + 14 = x + 2
3. 10 − 5x = x + 2
4. Add 5x and subtract 2 from both sides: 8 = 6x
5. Divide by 6: x = 4/3

Answerx = 4/3

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1. At each box, solve the equation, then leave along the arrow whose number equals your answer.
2. START: 8x = 20 + 3x, so 5x = 20, so x = 4. Take arrow 4 down to 2x − 9 = −3.
3. 2x − 9 = −3, so 2x = 6, so x = 3. Take arrow 3 to −2x = −42.
4. −2x = −42, so x = 21. Take arrow 21 up to 15 = 19 − 4x.
5. 15 = 19 − 4x, so 4x = 4, so x = 1. Take arrow 1 down to 2x + 3 = x + 5.
6. 2x + 3 = x + 5, so x = 2. Take arrow 2 down to 2x + 5 = 3(x − 1).
7. 2x + 5 = 3x − 3, so x = 8. Take arrow 8 to 2(x + 1) − 10 = 18.
8. 2(x + 1) − 10 = 18, so 2x − 8 = 18, so 2x = 26, so x = 13. Take arrow 13 to 8m + 8 = −72.
9. 8m + 8 = −72, so 8m = −80, so m = −10. Take arrow −10 down to −4 = 16 − 5k.
10. −4 = 16 − 5k, so 5k = 20, so k = 4. Take arrow +4 to 2x − 9 = 3 − x.
11. 2x − 9 = 3 − x, so 3x = 12, so x = 4. Take arrow 4 to 30 = 4 − 50n.
12. 30 = 4 − 50n, so 50n = −26, so n = −13/25. This box is next to END.

AnswerThe arrows on the path are 4, 3, 21, 1, 2, 8, 13, −10, 4, 4, ending at 30 = 4 − 50n (n = −13/25) and then END.

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1. Every child or donkey has exactly one head, so donkeys + children = 28.
2. Let the number of donkeys be d; then children = 28 − d.
3. Each donkey has 4 feet and each child has 2 feet: 4d + 2(28 − d) = 80.
4. 4d + 56 − 2d = 80.
5. 2d + 56 = 80, so 2d = 24.
6. Divide both sides by 2: d = 12.
7. Children = 28 − 12 = 16.

Answer12 donkeys and 16 children

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