Class 7 Ganita Prakash Chapter 15 Finding the Unknown Solutions

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Class 7 Ganita Prakash · Chapter 15 (Part 2 · Ch 7)
Finding the Unknown

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✔ Understood 0/76

🗝️ Key ideas of this chapter

• An equation says two expressions are equal (LHS = RHS). Solving = finding the value that makes it true.
• Do the same thing to both sides (add, subtract, multiply, divide by a non-zero number) — the balance stays.
• Shortcut: a term moves to the other side with the opposite sign; a factor moves as division; a divisor moves as multiplication.
• Always check by substituting the answer.
• Brahmagupta: Ax + B = Cx + D gives x = (D − B) ÷ (A − C).

📖 Textbook Page 164

📖 Page 164Fig. 7.1

The number at the top is the total weight and the mobile hangs level. Leaf = 3. Find the other weights.

1Level mobile → each side carries half of 16 = 8.

2Left: 3 + purple + 3 = 8 → purple = 2.

3Right: flower = 8.

✅ Answer: purple = 2, flower = 8

📖 Page 164Fig. 7.2

Total 24. Starfish = 2. Left: starfish, fish, starfish. Right: fish, grey fish. Find the fish weights.

1Each side = 12.

2Left: 2 + fish + 2 = 12 → fish = 8.

3Right: 8 + grey fish = 12 → grey fish = 4.

✅ Answer: fish = 8, grey fish = 4

📖 Page 164Fig. 7.3

Total 8. A book hangs on one side; two equal bags hang from a small bar on the other side.

1Each side = 4 → book = 4.

2Two equal bags make 4 → each bag = 2.

✅ Answer: book = 4, bag = 2

📖 Textbook Page 165

📖 Page 165Fig. 7.4

Total 18. Left: a sun (5) and 4 clouds; right: 3 lightning bolts.

1Each side = 9.

2Left: 5 + 4 clouds = 9 → 4 clouds = 4 → cloud = 1.

3Right: 3 bolts = 9 → bolt = 3.

✅ Answer: cloud = 1, bolt = 3

📖 Page 165Fig. 7.5

Total 40. Left: a crown and 5 gems; right: 4 crowns.

1Each side = 20.

2Right: 4 crowns = 20 → crown = 5.

3Left: 5 + 5 gems = 20 → 5 gems = 15 → gem = 3.

✅ Answer: crown = 5, gem = 3

📖 Page 165Fig. 7.6

Three bread slices (2 each) balance two fried eggs. Find an egg’s weight.

1Left = 2 + 2 + 2 = 6.

2Equation: 2e = 6

3e = 3

✅ Answer: egg = 3

📖 Page 165Fig. 7.7

Four crosses (4 each) balance two rings and a cross. Find a ring’s weight.

1Left = 16.

2Equation: 4 + 2y = 16

32y = 12 → y = 6

✅ Answer: ring = 6

📖 Page 165Fig. 7.8

A watermelon (10) balances two bananas and an orange (4). Find a banana’s weight.

1Equation: 2b + 4 = 10

22b = 6 → b = 3

✅ Answer: banana = 3

📖 Page 165Fig. 7.9

A sack and a 2 kg weight balance 10 kg and 2 kg. Find the sack.

1Remove 2 kg from both plates.

2s = 10

✅ Answer: 10 kg

📖 Page 165Fig. 7.10

Two equal sacks balance a 10 kg weight, a 4 kg weight and one sack. Find a sack.

1Remove one sack from each plate.

2One sack = 10 + 4 = 14

✅ Answer: 14 kg

📖 Textbook Page 166

📖 Page 166Fig. 7.11

Five sacks balance two sacks with 10 kg, 10 kg and 1 kg. Find a sack.

1Remove 2 sacks from each plate: 3 sacks = 21 kg.

21 sack = 7 kg

3Equation: 5s = 2s + 21

✅ Answer: 7 kg

📖 Page 166Fig. 7.12

90 sacks and a 50 kg weight balance 60 sacks and a 500 kg weight. Find a sack.

1Remove 60 sacks from each side: 30 sacks + 50 = 500.

2Remove 50 kg: 30 sacks = 450 kg.

31 sack = 15 kg

4Equation: 90s + 50 = 60s + 500

✅ Answer: 15 kg

📖 Textbook Page 167

📖 Page 167Matchstick pattern

Position n uses 2n + 1 sticks. Which position uses exactly 99 sticks? Can 200 sticks make one?

Positions 1 to 4: each new triangle needs 2 more sticks.

12n + 1 = 99 → 2n = 98 → n = 49.

22n + 1 = 200 → 2n = 199 → n = 99.5 — not a whole number.

3Every position uses an odd number of sticks, and 200 is even.

✅ Answer: Position 49; 200 sticks — not possible

📖 Textbook Page 168

📖 Page 168Framing equations

Frame equations for Figs. 7.6 to 7.11 and solve them.

17.6: 2e = 6 → e = 3. 7.7: 4 + 2y = 16 → y = 6. 7.8: 2b + 4 = 10 → b = 3.

27.9: s + 2 = 12 → s = 10. 7.10: 2s = s + 14 → s = 14. 7.11: 5s = 2s + 21 → s = 7.

✅ Answer: Same answers as before

📖 Page 168Trial and error

Solve 2n + 1 = 99 and 5x − 4 = 7 by trial and error. Can 2n + 1 = 99 have another solution?

1n = 40 → 81, n = 50 → 101, n = 49 → 99 ✔

2As n grows by 1, 2n + 1 grows by 2, so only n = 49 works.

35x − 4 = 7: x = 2 → 6, x = 3 → 11; between them; try x = 2.2 → 11 − 4 = 7 ✔ (trial and error is slow here!)

✅ Answer: n = 49 only; x = \(\dfrac{11}{5}\) = 2.2

📖 Textbook Page 169

📖 Page 169Balance property

15 + 8 = 23. Add, subtract, multiply or divide both sides by the same number. Is it still equal?

1Add 10: 33 = 33 ✔. Subtract 3: 20 = 20 ✔.

2× 2: 46 = 46 ✔. ÷ 23: 1 = 1 ✔ (never divide by 0).

3Doing the same to both sides keeps the balance.

✅ Answer: Yes, always

📖 Page 169Examples 1 and 2

14593 − 1459 + 145 − 14 + 88 = 13353. Find 14593 − 1459 + 145 − 14. Also 23 × 41 × 11 × 8 × 7 = 5,80,888; find 23 × 41 × 11 × 8.

1Subtract 88 from both sides: 13353 − 88 = 13265.

2Divide both sides by 7: 580888 ÷ 7 = 82984.

✅ Answer: 13265 and 82984

📖 Textbook Page 170

📖 Page 170Example 3

12345 − 5432 + 135 − 24 − (−67) = 7091. Find 12345 − 5432 + 135 − 24.

1Add (−67) to both sides to remove − (−67).

27091 + (−67) = 7024

3(The book’s working writes 132 instead of 135 — the answer 7024 is correct for 135.)

✅ Answer: 7024

📖 Page 170Example 4

(\(\dfrac{35}{113}\)) × 24 × 14 × (\(\dfrac{8}{9}\)) = \(\dfrac{94080}{1017}\). Find (\(\dfrac{35}{113}\)) × 24 × 14.

1Divide both sides by \(\dfrac{8}{9}\), i.e. multiply by \(\dfrac{9}{8}\).

2\(\dfrac{94080}{1017}\) × \(\dfrac{9}{8}\) = \(\dfrac{11760}{113}\)

✅ Answer: \(\dfrac{11760}{113}\)

📖 Textbook Page 171

📖 Page 1715x − 4 = 7

Solve 5x − 4 = 7 systematically and check.

1Add 4 to both sides: 5x = 11.

2Divide both sides by 5: x = \(\dfrac{11}{5}\).

3Check: 5 × \(\dfrac{11}{5}\) − 4 = 11 − 4 = 7 ✔

✅ Answer: x = \(\dfrac{11}{5}\)

📖 Page 171Example 5

Solve 11y + (−5) = 61 and check.

1Subtract (−5) from both sides: 11y = 66.

2y = 6

3Check: 66 − 5 = 61 ✔

✅ Answer: y = 6

📖 Textbook Page 172

📖 Page 172Example 6

Solve 6y + 7 = 4y + 21.

1Subtract 4y: 2y + 7 = 21.

2Subtract 7: 2y = 14.

3y = 7. Check: 49 = 49 ✔

✅ Answer: y = 7

📖 Page 172Figure it Out · Q1 (a)

Solve 3x − 10 = 35 and check.

1Add 10: 3x = 45

2x = 15

3Check: 45 − 10 = 35 ✔

✅ Answer: x = 15

📖 Page 172Figure it Out · Q1 (b)

Solve 5s = 3s and check.

1Subtract 3s: 2s = 0

2s = 0

3Check: 0 = 0 ✔

✅ Answer: s = 0

📖 Page 172Figure it Out · Q1 (c)

Solve 3u − 7 = 2u + 3 and check.

1Subtract 2u: u − 7 = 3

2u = 10

3Check: 23 = 23 ✔

✅ Answer: u = 10

📖 Page 172Figure it Out · Q1 (d)

Solve 4(m + 6) − 8 = 2m − 4 and check.

14m + 24 − 8 = 2m − 4 → 4m + 16 = 2m − 4

22m = −20 → m = −10

3Check: 4(−4) − 8 = −24 and 2(−10) − 4 = −24 ✔

✅ Answer: m = −10

📖 Page 172Figure it Out · Q1 (e)

Solve \(\dfrac{u}{15}\) = 6 and check.

1Multiply both sides by 15: u = 90

2Check: 90 ÷ 15 = 6 ✔

✅ Answer: u = 90

📖 Page 172Figure it Out · Q2

Frame an equation that has no solution.

1x + 4 = x + 5: subtracting x gives 4 = 5, which is never true.

2So no value of x works.

✅ Answer: e.g. x + 4 = x + 5

📖 Textbook Page 173

📖 Page 173Shortcut

Show the short way of solving 11y + (−5) = 61 and u ÷ 15 = 6.

1A term added on one side moves to the other side as its additive inverse: 11y = 61 − (−5) = 66.

2A factor on one side moves as division: y = 66 ÷ 11 = 6.

3A divisor moves as multiplication: u = 6 × 15 = 90.

✅ Answer: y = 6; u = 90

📖 Textbook Page 174

📖 Page 174Example 7 · Tiles

Ranjana’s step k uses 3k + 1 tiles. Can she use exactly 100 tiles? Which step?

Arms of k tiles on three sides + 1 centre tile = 3k + 1

13k + 1 = 100 → 3k = 99 → k = 33.

2Check: 3 × 33 + 1 = 100 ✔

✅ Answer: Yes — step 33

📖 Textbook Page 175

📖 Page 175Example 8 · Party

Snacks cost ₹25 a plate, delivery ₹50, budget ₹500, family of 5. How many friends can Madhubanti invite?

1Mahesh: 25p + 50 = 500 → 25p = 450 → p = 18 people.

2Srikanth: 25(f + 5) = 450 → f + 5 = 18 → f = 13.

3Both give 13 friends.

✅ Answer: 13 friends

📖 Textbook Page 177

📖 Page 177Example 9 · Savings

Jahnavi: ₹4000 + ₹650 a month. Sunita: ₹5050 + ₹500 a month. After how many months are they equal?

14000 + 650m = 5050 + 500m

2150m = 1050 → m = 7

3Check: 4000 + 4550 = 8550 and 5050 + 3500 = 8550 ✔

✅ Answer: 7 months

📖 Page 177Example 10

Solve 28(x + 4) + 300 = 1000 in two ways.

1Way 1: 28(x + 4) = 700 → x + 4 = 25 → x = 21.

2Way 2: 28x + 112 + 300 = 1000 → 28x = 588 → x = 21.

✅ Answer: x = 21

📖 Textbook Page 178

📖 Page 178Example 11 · Riyaz’s trick

Think of a number, subtract 3, multiply by 4, add 8. The answer is 24. What was the number? Give a quick rule.

1Expression: 4(x − 3) + 8 = 4x − 4.

24x − 4 = 24 → 4x = 28 → x = 7.

3Rule: starting number = (answer ÷ 4) + 1.

✅ Answer: 7; start = answer ÷ 4 + 1

📖 Textbook Page 179

📖 Page 179Example 12 · Marbles

Ramesh and Suresh have 60 marbles; Ramesh has 30 more. How many each?

1Suresh y, Ramesh y + 30: 2y + 30 = 60.

22y = 30 → y = 15.

3Suresh 15, Ramesh 45.

✅ Answer: Suresh 15, Ramesh 45

📖 Textbook Page 180

📖 Page 180Generating equations

Write equations whose solution is y = 5. Can you reverse the chain y + 1 = 6 → −y − 1 = −6 → −1 = −6 + y → 5 = y?

1Examples: y + 1 = 6, 3y = 15, 2y − 4 = 6, y ÷ 5 = 1, 10 − y = 5.

2Reverse: 5 = y → subtract 6: −1 = −6 + y → subtract y: −y − 1 = −6 → multiply by −1: y + 1 = 6.

3Each reverse step undoes the forward step (add ↔ subtract, × ↔ ÷).

✅ Answer: e.g. 3y = 15; reverse each operation

📖 Page 180Example 13

Give a real-life situation for 100x + 75 = 250.

1100 plates of snacks at ₹x each plus ₹75 delivery cost ₹250 in all.

2100x = 175 → x = 1.75

✅ Answer: e.g. 100 items at ₹x plus ₹75 = ₹250 → x = ₹1.75

📖 Textbook Page 181

📖 Page 181Figure it Out · Q1

Write 5 equations whose solution is x = −2.

1Start from x = −2 and do the same thing to both sides.

✅ Answer: x + 5 = 3; 2x = −4; 3x + 1 = −5; \(\dfrac{x}{2}\) = −1; 4 − x = 6

📖 Page 181Figure it Out · Q2

Find each unknown: (a) 2y = 60; (b) −8 = 5x − 3; (c) −53w = −15; (d) 13 − z = 8; (e) k + 8 = 12 − k; (f) 7m = m − 3; (g) 3n = 10 + n

1(a) 60 ÷ 2 (b) −5 = 5x → x = −1 (c) w = −15 ÷ −53

2(d) 13 − 8 = z (e) 2k = 4 (f) 6m = −3 (g) 2n = 10

✅ Answer: (a) y = 30; (b) x = −1; (c) w = \(\dfrac{15}{53}\); (d) z = 5; (e) k = 2; (f) m = \(-\dfrac{1}{2}\); (g) n = 5

📖 Page 181Figure it Out · Q3

A 3-digit number: hundreds digit is 3 less than tens; tens is 3 less than units; digit sum 15. Find it.

1Units u, tens u − 3, hundreds u − 6.

2(u − 6) + (u − 3) + u = 15 → 3u = 24 → u = 8.

3Digits 2, 5, 8.

✅ Answer: 258

📖 Page 181Figure it Out · Q4

A brick weighs 1 kg more than half its weight. Find its weight.

1b = \(\dfrac{b}{2}\) + 1 → \(\dfrac{b}{2}\) = 1

2b = 2

✅ Answer: 2 kg

📖 Page 181Figure it Out · Q5

One quarter of a number increased by 9 gives the number. Find it.

1\(\dfrac{n}{4}\) + 9 = n → 9 = \(\dfrac{3n}{4}\)

2n = 12

✅ Answer: 12

📖 Page 181Figure it Out · Q6

Given 4k + 1 = 13, find (a) 8k + 2 (b) 4k (c) k (d) 4k − 1 (e) −k − 2.

1(a) 2 × 13 = 26

2(b) 13 − 1 = 12

3(c) 12 ÷ 4 = 3

4(d) 12 − 1 = 11

5(e) −3 − 2 = −5

✅ Answer: (a) 26 (b) 12 (c) 3 (d) 11 (e) −5

📖 Textbook Page 182

📖 Page 182Mind the Mistake 1

Check the steps: 4x + 6 = 10 → 4x = 10 + 6 → x = 4. Find and correct the mistake.

16 should be subtracted: 4x = 10 − 6 = 4

2Correct answer: x = 1

✅ Answer: x = 1

📖 Page 182Mind the Mistake 2

Check the steps: 7 − 8z = 5 → 8z = 7 − 5 = 2 → z = 4. Find and correct the mistake.

18z = 2 gives z = 2 ÷ 8, not 4

2Correct answer: z = \(\dfrac{1}{4}\)

✅ Answer: z = \(\dfrac{1}{4}\)

📖 Page 182Mind the Mistake 3

Check the steps: 2v − 4 = 6 → v − 4 = 6 − 2 → v = 8. Find and correct the mistake.

1Dividing by 2 must divide every term; first add 4: 2v = 10

2Correct answer: v = 5

✅ Answer: v = 5

📖 Page 182Mind the Mistake 4

Check the steps: 5z + 2 = 3z − 4 → 5z + 3z = −4 + 2. Find and correct the mistake.

1Moving 3z gives 5z − 3z; moving 2 gives −4 − 2: 2z = −6

2Correct answer: z = −3

✅ Answer: z = −3

📖 Page 182Mind the Mistake 5

Check the steps: 15w − 4w = 26 → 15w = 26 + 4w → 15w = 30. Find and correct the mistake.

126 + 4w is not 30; simply 11w = 26

2Correct answer: w = \(\dfrac{26}{11}\)

✅ Answer: w = \(\dfrac{26}{11}\)

📖 Page 182Mind the Mistake 6

Check the steps: 3x + 1 = −12 → x + 1 = −12 ÷ 3. Find and correct the mistake.

1The 1 was not divided; first 3x = −13

2Correct answer: x = \(-\dfrac{13}{3}\)

✅ Answer: x = \(-\dfrac{13}{3}\)

📖 Page 182Mind the Mistake 7

Check the steps: 4(4q + 2) = 50 → 4(4q) = 50 − 2. Find and correct the mistake.

1The 2 is also multiplied by 4: 16q + 8 = 50 → 16q = 42

2Correct answer: q = \(\dfrac{21}{8}\)

✅ Answer: q = \(\dfrac{21}{8}\)

📖 Page 182Mind the Mistake 8

Check the steps: −2(3 − 4x) = 14 → −6 − 8x = 14. Find and correct the mistake.

1−2 × (−4x) = +8x: −6 + 8x = 14 → 8x = 20

2Correct answer: x = \(\dfrac{5}{2}\)

✅ Answer: x = \(\dfrac{5}{2}\)

📖 Page 182Mind the Mistake 9

Check the steps: 3(7y + 4) = 9 + 5y → 7y + 4 = 3 + 5y. Find and correct the mistake.

15y was not divided by 3. Expand: 21y + 12 = 9 + 5y → 16y = −3

2Correct answer: y = \(-\dfrac{3}{16}\)

✅ Answer: y = \(-\dfrac{3}{16}\)

📖 Textbook Page 183

📖 Page 183Example 16 · Horses

Bhāskarāchārya: One man has ₹300 and 6 horses; another has 10 horses and a debt of ₹100. They are equally rich. Price of a horse?

1300 + 6x = 10x − 100

2400 = 4x → x = 100

3Check: 300 + 600 = 900 and 1000 − 100 = 900 ✔

✅ Answer: ₹100

📖 Textbook Page 184

📖 Page 184Brahmagupta’s formula

For Ax + B = Cx + D, x = (D − B) ÷ (A − C). Solve 5x + 4 = 3x + 8, 3x − 6 = 2x + 4 and 2x + 3 = 4x + 5.

15x + 4 = 3x + 8: x = (8 − 4) ÷ (5 − 3) = 2

23x − 6 = 2x + 4: x = (4 − (−6)) ÷ (3 − 2) = 10

32x + 3 = 4x + 5: x = (5 − 3) ÷ (2 − 4) = −1

✅ Answer: 2, 10, −1

📖 Textbook Page 185

📖 Page 185Figure it Out · Q1

Fill in with integers: (a) 5 × __ − 8 = 37 (b) 37 − (33 − __) = 35 (c) −3 × (−11 + __) = 45

1(a) 5x = 45 → 9

2(b) 33 − x = 2 → x = 31

3(c) −11 + x = −15 → x = −4

✅ Answer: (a) 9 (b) 31 (c) −4

📖 Page 185Figure it Out · Q2

Ranju earns ₹750 in equal numbers of ₹50 and ₹100 notes. How many of each?

150n + 100n = 750 → 150n = 750

2n = 5

✅ Answer: 5 of each

📖 Page 185Figure it Out · Q3

Each black blob hides the same number of dots. 25 dots in all. How many under each blob? Write the equation.

14 dots can be seen and there are 3 blobs.

23b + 4 = 25 → 3b = 21

3b = 7

✅ Answer: 7 dots; 3b + 4 = 25

📖 Page 185Figure it Out · Q4 (a)

Machine: input → +3 → ×4 → −5. Find the inputs for outputs 43 and 75.

1Equation: 4(x + 3) − 5 = output.

243: 4(x + 3) = 48 → x + 3 = 12 → x = 9

375: 4(x + 3) = 80 → x + 3 = 20 → x = 17

✅ Answer: 9 and 17

📖 Textbook Page 186

📖 Page 186Figure it Out · Q4 (b)

Machine: (input × 3) − (input + 3). Example 12 → 36 − 15 = 21. Find the inputs for 63 and 227.

13x − (x + 3) = 2x − 3

22x − 3 = 63 → x = 33

32x − 3 = 227 → x = 115

✅ Answer: 33 and 115

📖 Page 186Figure it Out · Q5

Find the inputs: (a) ÷3 then ÷3 gives +5 (b) −4 then −4 gives −11.

1(a) x ÷ 9 = 5 → x = 45

2(b) x − 8 = −11 → x = −3

✅ Answer: (a) 45 (b) −3

📖 Page 186Figure it Out · Q6

A taxi charges ₹800 per day plus ₹20 per km. The bill is ₹2200. How many km?

1800 + 20k = 2200 → 20k = 1400

2k = 70

✅ Answer: 70 km

📖 Textbook Page 187

📖 Page 187Figure it Out · Q7

Two numbers add to 76; one is three times the other. Find them.

1n + 3n = 76 → 4n = 76 → n = 19

2Other = 57

✅ Answer: 19 and 57

📖 Page 187Figure it Out · Q8

The window is 34 cm tall: a 3 cm frame at the top (and bottom), 2 cm rods and equal gaps. Find the gap between two rods.

1Count from the picture: 5 rods and 6 equal gaps; frames of 3 cm at top and bottom.

23 + 3 + 5 × 2 + 6g = 34 → 16 + 6g = 34

36g = 18 → g = 3

✅ Answer: 3 cm

📖 Page 187Figure it Out · Q9

A juice costs ₹15 less than a milkshake. 4 juices and 7 milkshakes cost ₹600. Find each price.

1Milkshake m, juice m − 15: 4(m − 15) + 7m = 600

211m − 60 = 600 → m = 60

3Juice = 45. Check: 180 + 420 = 600 ✔

✅ Answer: Juice ₹45, milkshake ₹60

📖 Page 187Figure it Out · Q10

Given 28p − 36 = 98, find 14p − 19 and 28p − 38.

128p = 134, so 14p = 67.

214p − 19 = 48

328p − 38 = 134 − 38 = 96

✅ Answer: 48 and 96

📖 Page 187Figure it Out · Q11

Find the mistakes: (a) 6x + 9 = 66 → x + 9 = 11 (b) 14y + 24 = 36 → 7y + 12 = 18 → 7y = 6 → y = \(\dfrac{6}{7}\) (c) 4x − 5 = 9x + 8 → 4x = 9x + 8 − 5 → … → x = \(-\dfrac{5}{3}\)

1(a) Only 6x and 66 were divided — 9 was not. Correct: 6x = 57 → x = \(\dfrac{19}{2}\).

2(b) Correct! Every term was divided by 2: y = \(\dfrac{6}{7}\).

3(c) −5 should move as +5: 4x = 9x + 13 → −5x = 13 → x = \(-\dfrac{13}{5}\) (also the last step was turned upside down).

✅ Answer: (a) x = \(\dfrac{19}{2}\) (b) correct (c) x = \(-\dfrac{13}{5}\)

📖 Page 187Figure it Out · Q12

Find the angles of the triangles.

1Isosceles: y + (y + 15) + (y + 15) = 180 → 3y = 150 → y = 50 → angles 50°, 65°, 65°.

2Second: x + (x − 10) + (x + 10) = 180 → 3x = 180 → x = 60 → angles 60°, 50°, 70°.

✅ Answer: 50°, 65°, 65° and 60°, 50°, 70°

📖 Textbook Page 188

📖 Page 188Figure it Out · Q13

Write 4 equations whose solution is u = 6.

1Do the same operation to both sides of u = 6.

✅ Answer: u + 4 = 10; 2u = 12; 3u − 5 = 13; \(\dfrac{u}{3}\) = 2

📖 Page 188Figure it Out · Q14

Bakhśhālī: the 2nd gets twice the 1st, the 3rd thrice the 2nd, the 4th four times the 3rd. Total 132. Find the 1st person’s share.

1Shares: x, 2x, 6x, 24x.

233x = 132 → x = 4

3Check: 4 + 8 + 24 + 96 = 132 ✔

✅ Answer: 4

📖 Page 188Figure it Out · Q15

A giraffe’s height is 2.5 m more than half its height. How tall is it?

1h = \(\dfrac{h}{2}\) + 2.5 → \(\dfrac{h}{2}\) = 2.5

2h = 5

✅ Answer: 5 m

📖 Page 188Figure it Out · Q16

Two stick sequences: (a) squares in position 11 (b) sticks in position 11 (c) 85 sticks? (d) 150 sticks?

Sequence 1 (top) and sequence 2 (bottom): positions 1, 2, 3

1Sequence 1 (n squares + a triangle): squares n, sticks 3n + 1 + 2 = 3n + 3.

2Position 11: 11 squares, 36 sticks. 3n + 3 = 85 → n not whole → no. 3n + 3 = 150 → n = 49 → yes.

3Sequence 2 (staircase): squares 3n + 1, sticks 9n + 4 (position 1: 4 squares, 13 sticks; each step adds 3 squares, 9 sticks).

4Position 11: 34 squares, 103 sticks. 9n + 4 = 85 → n = 9 → yes. 9n + 4 = 150 → n not whole → no.

✅ Answer: Seq 1: 11, 36, no, position 49. Seq 2: 34, 103, position 9, no

📖 Page 188Figure it Out · Q17

A number increased by 36 equals ten times itself. Find it.

1n + 36 = 10n → 9n = 36

2n = 4

✅ Answer: 4

📖 Page 188Figure it Out · Q18

Solve: (a) 5(r + 2) = 10; (b) −3(u + 2) = 2(u − 1); (c) 2(7 − 2n) = −6; (d) 2(x − 4) = −16; (e) 6(x − 1) = 2(x − 1) − 4; (f) 3 − 7s = 7 − 3s; (g) 2x + 1 = 6 − (2x − 3); (h) 10 − 5x = 3(x − 4) − 2(x − 7)

1(a) r + 2 = 2 (b) −3u − 6 = 2u − 2 → −5u = 4 (c) 7 − 2n = −3 → 2n = 10 (d) x − 4 = −8

2(e) 6x − 6 = 2x − 6 → 4x = 0 (f) −4 = 4s (g) 2x + 1 = 9 − 2x → 4x = 8 (h) 10 − 5x = x + 2 → 6x = 8

✅ Answer: (a) r = 0; (b) u = \(-\dfrac{4}{5}\); (c) n = 5; (d) x = −4; (e) x = 0; (f) s = −1; (g) x = 2; (h) x = \(\dfrac{4}{3}\)

📖 Textbook Page 189

📖 Page 189Figure it Out · Q19

Solve the equations to find the path from Start to End (each arrow shows the answer of the box it leaves).

1Start: 8x = 20 + 3x → x = 4 → go to 2x − 9 = −3 → x = 3.

2→ −2x = −42 → x = 21 → 15 = 19 − 4x → x = 1 → 2x + 3 = x + 5 → x = 2.

3→ 2x + 5 = 3(x − 1) → x = 8 → 2(x + 1) − 10 = 18 → x = 13 → 8m + 8 = −72 → m = −10.

4→ −4 = 16 − 5k → k = 4 → 2x − 9 = 3 − x → x = 4 → 30 = 4 − 50n → n = \(-\dfrac{13}{25}\) → END.

5(The box −7 = 11 − 3x, with answer 6, is not on the path.)

✅ Answer: See the coloured path

📖 Page 189Figure it Out · Q20

Children and donkeys have 28 heads and 80 feet. How many of each?

1Donkeys d, children 28 − d: 4d + 2(28 − d) = 80.

22d + 56 = 80 → d = 12

3Children 16. Check: 48 + 32 = 80 ✔

✅ Answer: 12 donkeys, 16 children

📖 Textbook Page 190

📖 Page 190A Magic Trick

Think of a number, ×2, +10, ÷2, − the number, +3. Why is the answer always 8?

1x → 2x → 2x + 10 → x + 5 → 5 → 8.

2The starting number cancels out, so everyone gets 8.

✅ Answer: Always 8 — x cancels

🌍 Where do we use this?

• Bills, savings, taxi fares, recipes, puzzles and number tricks.

🚀 Link to higher classes

• Class 8: linear equations in one variable; Class 9–10: equations in two variables.

⬆ Top 🏠 Class 7 Hub

🔑 keytoenjoylearningmaths.blogspot.com · Solutions written in our own words, based on NCERT Ganita Prakash Class 7 (Part 2)