7. Finding the Unknown
Answers to all Figure it Out and in-text questions on equations and finding unknowns. CBSE / NCERT Class 7 Ganita Prakash Part 2.
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7.1 Find the Unknowns
In a hanging balance, both sides have equal weight. So, each side has half of the total.
- Fig. 7.1 (16): each side = 8. Left: 3 + bud + 3 = 8, so bud = 2. Right: flower = 8.
- Fig. 7.2 (24): each side = 12. Left: 2 + fish + 2 = 12, so fish = 8. Right: 8 + submarine = 12, so submarine = 4.
- Fig. 7.3 (8): each side = 4. Book = 4. Two boxes = 4, so each box = 2.
- Fig. 7.4 (18): each side = 9. Left: 5 + 4 clouds = 9, so cloud = 1. Right: 3 lightning bolts = 9, so bolt = 3.
- Fig. 7.5 (40): each side = 20. Right: 4 crowns = 20, so crown = 5. Left: 5 + 5 gems = 20, so gem = 3.
- Fig. 7.6: 3 slices × 2 = 6 = 2 eggs. So, egg = 3.
- Fig. 7.7: 4 crosses × 4 = 16 = 4 + 2 rings. So, 2 rings = 12, and ring = 6.
- Fig. 7.8: 10 = banana + 4 + banana. So, 2 bananas = 6, and banana = 3.
- Fig. 7.9: sack + 2 = 10 + 2. Remove 2 kg from both plates. Sack = 10 kg.
- Fig. 7.10: 2 sacks = sack + 10 + 4. Remove one sack from both plates. Sack = 14 kg.
- Fig. 7.11: 5 sacks = 2 sacks + 10 + 10 + 1. Remove 2 sacks from both plates: 3 sacks = 21. Sack = 7 kg.
- Fig. 7.12: 90 sacks + 50 = 60 sacks + 500. Remove 60 sacks and 50 kg from both plates: 30 sacks = 450. Sack = 15 kg.
2n + 1 = 99. Take away 1: 2n = 98. Half of 98 is 49.
So, n = 49. Jasmine’s arrangement is at position 49.
No. 2n + 1 is always an odd number. 200 is even.
(2n + 1 = 200 gives n = 99.5, which is not a position number.)
- Fig. 7.6: 2e = 6, so e = 3.
- Fig. 7.7: 4 + 2y = 16, so 2y = 12 and y = 6.
- Fig. 7.8: 2b + 4 = 10, so 2b = 6 and b = 3.
- Fig. 7.9: s + 2 = 10 + 2, so s = 10.
- Fig. 7.10: 2s = s + 14, so s = 14.
- Fig. 7.11: 5s = 2s + 21, so 3s = 21 and s = 7.
We get the same values as before.
Examples: x + 7 = 12 (x = 5); 3y = 24 (y = 8); 2a − 1 = 9 (a = 5); p ÷ 4 = 3 (p = 12); 5m + 2 = 3m + 10 (m = 4).
7.2 Solving Equations Systematically
No. As n increases, 2n + 1 also increases. So, only one value, n = 49, gives 99.
x = 2 gives 6. x = 3 gives 11. So, x is between 2 and 3. Trial and error is slow here.
Systematically: 5x = 7 + 4 = 11, so x = 11/5.
Check: 5 × 11/5 − 4 = 11 − 4 = 7 = RHS.
LHS = 15 + 8 + 10 = 33. RHS = 23 + 10 = 33. The equality still holds.
Yes. 23 × 41 × 11 × 8 × 7 ÷ 7 = 5,80,888 ÷ 7.
So, 23 × 41 × 11 × 8 = 82,984.
LHS = 11 × 6 + (−5) = 66 − 5 = 61 = RHS. So, y = 6 is correct.
1. Solve and check.
(a) 3x − 10 = 35 → 3x = 45 → x = 15. Check: 45 − 10 = 35.
(b) 5s = 3s → 5s − 3s = 0 → 2s = 0 → s = 0. Check: 0 = 0.
(c) 3u − 7 = 2u + 3 → 3u − 2u = 3 + 7 → u = 10. Check: 23 = 23.
(d) 4(m + 6) − 8 = 2m − 4 → 4m + 16 = 2m − 4 → 2m = −20 → m = −10. Check: 4 × (−4) − 8 = −24 and 2 × (−10) − 4 = −24.
(e) u ÷ 15 = 6 → u = 6 × 15 → u = 90. Check: 90 ÷ 15 = 6.
2. Frame an equation that has no solution.
x + 4 = x + 5. For any x, the LHS is always 1 less than the RHS. So, it has no solution.
3k = 100 − 1 = 99. k = 99 ÷ 3 = 33.
Yes, Ranjana can make it. It will be Step 33.
Jahnavi: 4000 + 650 × 7 = 4000 + 4550 = ₹8550.
Sunita: 5050 + 500 × 7 = 5050 + 3500 = ₹8550. Both are equal.
Start with 10: 10 − 3 = 7, 7 × 4 = 28, 28 + 8 = 36. Start with 5: 2 × 4 + 8 = 16.
The final answer = 4 × (starting number − 1).
Rule: divide the final answer by 4, then add 1. For 24: 24 ÷ 4 + 1 = 7.
2y = 60 − 30 = 30. So, y = 15.
Suresh has 15 marbles. Ramesh has 15 + 30 = 45 marbles.
y + 3 = 8, 2y = 10, y − 7 = −2, 4y + 1 = 21, y ÷ 5 = 1.
Chain 1: 5 = y → subtract 6: −1 = −6 + y → subtract y: −1 − y = −6 → multiply by −1: y + 1 = 6.
Chain 2: y = 5 → add 2: y + 2 = 7 → multiply by 3: 3y + 6 = 21 → subtract 6: 3y = 15.
Going up, we use the inverse operations in the reverse order.
y = 5 in every equation. Doing the same operation on both sides does not change the solution.
1. Write 5 equations whose solution is x = −2.
x + 5 = 3; 3x = −6; x − 4 = −6; 2x + 7 = 3; 5x + 1 = 2x − 5.
2. Find the value of each unknown.
(a) 2y = 60 → y = 30.
(b) −8 = 5x − 3 → 5x = −5 → x = −1.
(c) −53w = −15 → w = 15/53.
(d) 13 − z = 8 → z = 5.
(e) k + 8 = 12 − k → 2k = 4 → k = 2.
(f) 7m = m − 3 → 6m = −3 → m = −1/2.
(g) 3n = 10 + n → 2n = 10 → n = 5.
In 2(c), if the equation is meant to be −(5/3)w = −15, then w = 9.
3. Who am I?
Let the units digit be u. Then the tens digit is u − 3, and the hundreds digit is u − 6.
u + (u − 3) + (u − 6) = 15 → 3u − 9 = 15 → 3u = 24 → u = 8.
Digits: 2, 5, 8. The number is 258.
4. Weight of a brick is 1 kg more than half its weight.
w = w/2 + 1 → w − w/2 = 1 → w/2 = 1 → w = 2 kg.
5. One quarter of a number increased by 9 gives the same number.
n/4 + 9 = n → 9 = n − n/4 = 3n/4 → n = 9 × 4 ÷ 3 = 12.
6. Given 4k + 1 = 13, find:
(a) 8k + 2 = 2 × (4k + 1) = 2 × 13 = 26
(b) 4k = 13 − 1 = 12
(c) k = 12 ÷ 4 = 3
(d) 4k − 1 = 12 − 1 = 11
(e) −k − 2 = −3 − 2 = −5
7.3 Mind the Mistake, Mend the Mistake
- 4x + 6 = 10 → 4x = 10 + 6 is wrong. +6 moves as −6. 4x = 10 − 6 = 4, so x = 1.
- 7 − 8z = 5 → 8z = 2 is right. But z = 4 is wrong. z = 2 ÷ 8, so z = 1/4.
- 2v − 4 = 6 → v − 4 = 6 − 2 is wrong. Correct: 2v = 6 + 4 = 10, so v = 5.
- 5z + 2 = 3z − 4 → 5z + 3z is wrong. 3z moves as −3z, and +2 moves as −2. 5z − 3z = −4 − 2 → 2z = −6 → z = −3.
- 15w = 26 + 4w is right. But 15w = 30 is wrong (26 and 4w cannot be added). 15w − 4w = 26 → 11w = 26 → w = 26/11.
- 3x + 1 = −12 → x + 1 = −12/3 is wrong. Only 3x was divided by 3. Correct: 3x = −13 → x = −13/3.
- 4(4q + 2) = 50 → 4(4q) = 50 − 2 is wrong. 4 × 2 = 8 must be taken away. 16q + 8 = 50 → 16q = 42 → q = 21/8.
- −2(3 − 4x) = 14 → −6v − 8x is wrong. −2 × 3 = −6 and −2 × (−4x) = +8x. −6 + 8x = 14 → 8x = 20 → x = 5/2.
- 3(7y + 4) = 9 + 5y → 7y + 4 = 9/3 + 5y is wrong. 5y was not divided by 3. Correct: 21y + 12 = 9 + 5y → 16y = −3 → y = −3/16. (Also, 2y = 4 − 3 should have been 2y = 3 − 4.)
7.4 A Pinch of History
For Ax + B = Cx + D, Brahmagupta’s formula is x = (D − B) ÷ (A − C).
5x + 4 = 3x + 8: x = (8 − 4) ÷ (5 − 3) = 4 ÷ 2 = 2.
3x − 6 = 2x + 4: x = (4 − (−6)) ÷ (3 − 2) = 10 ÷ 1 = 10.
x = (5 − 3) ÷ (2 − 4) = 2 ÷ (−2) = −1.
Check: 2(−1) + 3 = 1 and 4(−1) + 5 = 1.
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1. Fill in the blanks with integers.
(a) 5 × 9 − 8 = 37 (5 × __ = 45)
(b) 37 − (33 − 31) = 35 (33 − __ = 2)
(c) −3 × (−11 + (−4)) = 45 (−11 + __ = −15)
2. Ranju gets an equal number of ₹50 and ₹100 notes for ₹750.
Let the number of each note be n. 50n + 100n = 750 → 150n = 750 → n = 5.
She has 5 notes of ₹50 and 5 notes of ₹100.
3. Each blob hides an equal number of dots. Total = 25.
There are 3 blobs and 4 visible dots. Equation: 3b + 4 = 25.
3b = 21 → b = 7. Each blob covers 7 dots.
4. Find the inputs.
(a) Rule: (x + 3) × 4 − 5.
4(x + 3) − 5 = 43 → 4(x + 3) = 48 → x + 3 = 12 → x = 9.
4(x + 3) − 5 = 75 → 4(x + 3) = 80 → x + 3 = 20 → x = 17.
(b) Rule: 3x − (x + 3) = 2x − 3. (Check: 12 gives 36 − 15 = 21.)
2x − 3 = 63 → 2x = 66 → x = 33.
2x − 3 = 227 → 2x = 230 → x = 115.
5. Inputs to these machines.
x ÷ 3 ÷ 3 = 5 → x ÷ 9 = 5 → x = 45.
x − 4 − 4 = −11 → x − 8 = −11 → x = −3.
6. Taxi: ₹800 fixed plus ₹20 per km. Total ₹2200.
800 + 20k = 2200 → 20k = 1400 → k = 70 km.
7. Sum of two numbers is 76. One is three times the other.
x + 3x = 76 → 4x = 76 → x = 19. The numbers are 19 and 57.
8. Gap between two rods in the grill.
The window is 34 cm tall. The top and bottom frames are 3 cm each. There are 5 rods of 2 cm each and 6 equal gaps of g cm.
3 + 3 + 5 × 2 + 6g = 34 → 16 + 6g = 34 → 6g = 18 → g = 3 cm.
9. Fruit juice and milkshake.
Let a milkshake cost ₹m. A juice costs ₹(m − 15).
4(m − 15) + 7m = 600 → 11m − 60 = 600 → 11m = 660 → m = 60.
Milkshake = ₹60, fruit juice = ₹45.
10. Given 28p − 36 = 98, find 14p − 19 and 28p − 38.
28p = 134, so 14p = 67. Then 14p − 19 = 67 − 19 = 48.
28p − 38 = (28p − 36) − 2 = 98 − 2 = 96.
11. Identify and correct the mistakes.
(a) 6x + 9 = 66 → x + 9 = 11 is wrong. 9 was not divided by 6. Correct: 6x = 57 → x = 57/6 = 19/2.
(b) All the steps are correct. y = 6/7.
(c) 4x = 9x + 8 − 5 is wrong. −5 moves as +5: 4x = 9x + 13 → −5x = 13 → x = −13/5. (Also, x = −5/3 was written upside down.)
12. Find the angles of the triangles.
First triangle (isosceles): y + (y + 15) + (y + 15) = 180 → 3y + 30 = 180 → y = 50. Angles: 50°, 65°, 65°.
Second triangle: x + (x − 10) + (x + 10) = 180 → 3x = 180 → x = 60. Angles: 60°, 50°, 70°.
13. Write 4 equations whose solution is u = 6.
u + 4 = 10; 2u = 12; u ÷ 3 = 2; 3u − 8 = 10.
14. Bakhśhālī problem: total 132.
First: x. Second: 2x. Third: 3 × 2x = 6x. Fourth: 4 × 6x = 24x.
x + 2x + 6x + 24x = 132 → 33x = 132 → x = 4.
15. Height of the giraffe is 2½ m more than half its height.
h = h/2 + 2.5 → h/2 = 2.5 → h = 5 m.
16. Stick patterns.
Arrow pattern: position n has n squares and a triangle. Sticks = 3n + 1 (squares) + 2 (triangle) = 3n + 3.
(a) 11 squares. (b) 3 × 11 + 3 = 36 sticks. (c) 3n + 3 = 85 → 3n = 82. Not a whole number, so no. (d) 3n + 3 = 150 → 3n = 147 → yes, position 49.
Staircase pattern: positions 1, 2, 3, … have 4, 7, 10, … squares and 13, 22, 31, … sticks. Squares = 3n + 1. Sticks = 9n + 4.
(a) 3 × 11 + 1 = 34 squares. (b) 9 × 11 + 4 = 103 sticks. (c) 9n + 4 = 85 → 9n = 81 → yes, position 9. (d) 9n + 4 = 150 → 9n = 146. Not a whole number, so no.
17. A number increased by 36 is equal to ten times itself.
x + 36 = 10x → 9x = 36 → x = 4.
18. Solve these equations.
(a) 5(r + 2) = 10 → r + 2 = 2 → r = 0.
(b) −3(u + 2) = 2(u − 1) → −3u − 6 = 2u − 2 → −5u = 4 → u = −4/5.
(c) 2(7 − 2n) = −6 → 7 − 2n = −3 → 2n = 10 → n = 5.
(d) 2(x − 4) = −16 → x − 4 = −8 → x = −4.
(e) 6(x − 1) = 2(x − 1) − 4 → 6x − 6 = 2x − 6 → 4x = 0 → x = 0.
(f) 3 − 7s = 7 − 3s → −4s = 4 → s = −1.
(g) 2x + 1 = 6 − (2x − 3) → 2x + 1 = 9 − 2x → 4x = 8 → x = 2.
(h) 10 − 5x = 3(x − 4) − 2(x − 7) → 10 − 5x = x + 2 → 6x = 8 → x = 4/3.
19. Find a path from Start to End.
Solve each box and follow the arrow with that answer:
x = 4
x = 3
x = 21
x = 1
x = 2
x = 8
x = 13
m = −10
k = 4
x = 4
n = −13/25 → END
20. Children and donkeys: 28 heads and 80 feet.
Let the number of donkeys be d. Children = 28 − d.
4d + 2(28 − d) = 80 → 2d + 56 = 80 → 2d = 24 → d = 12.
12 donkeys and 16 children.
A Magic Trick
Let the number be x.
Multiply by 2: 2x. Add 10: 2x + 10. Divide by 2: x + 5. Subtract x: 5. Add 3: 8.
The x cancels out. So, the answer is always 8, whatever number you start with.