Question 99-sticks
A matchstick pattern has 2n+1 sticks at position n. Which position uses 99 sticks?
Steps only
2n+1=99. Subtract 1: 2n=98. Divide by 2: n=49.
Answer
n=49.
CLASS 7 · GANITA PRAKASH PART II · CHAPTER 7
Finding the Unknown
Understand equations as balanced statements and solve unknowns with clear inverse steps.
45 worked answersAll 45 end-of-chapter questions
What you will learn
- Represent unknown quantities with letters.
- Solve one-variable equations while preserving equality.
- Check solutions by substitution.
- Translate everyday situations into equations.
45 worked answers · short steps first, explanations when you need them
Figure it Out and chapter activities
Textbook pages 164–190
Question 200-sticks
Can the same pattern use exactly 200 sticks?
Steps only
An odd number plus 1 is even, so 2n+1 is odd for every whole-number position n.
Answer
No. 2n+1 is always odd, while 200 is even.
Question figure-it-out-1a
Solve 3x−10=35 and check.
Steps only
Add 10: 3x=45. Divide by 3: x=15. Check: 45−10=35.
Answer
x=15.
Question figure-it-out-1b
Solve 5s=3s and check.
Steps only
Subtract 3s from both sides: 2s=0. Divide by 2: s=0. Check: 0=0.
Answer
s=0.
Question figure-it-out-1c
Solve 3u−7=2u+3 and check.
Steps only
Subtract 2u: u−7=3. Add 7: u=10. Check: 30−7=20+3=23.
Answer
u=10.
Question figure-it-out-1d
Solve 4(m+6)−8=2m−4 and check.
Steps only
Expand: 4m+16=2m−4. Subtract 2m and 16: 2m=−20, so m=−10.
Answer
m=−10.
Question figure-it-out-1e
Solve u/15=6 and check.
Steps only
Multiply both sides by 15: u=90. Check: 90÷15=6.
Answer
u=90.
Question no-solution
Give an equation with no solution using “4 more than a number” and “5 more than a number.”
Steps only
Subtract x from both sides; this would require 4=5, which is false.
Answer
x+4=x+5 has no solution.
Question mistake-4x
Correct this work for 4x+6=10: 4x=10+6, x=4.
Steps only
Subtract 6 from both sides: 4x=4. Divide by 4: x=1. The shown work incorrectly added 6.
Answer
Correct solution: x=1.
Question mistake-7-8z
Solve 7−8z=5 and identify the sign carefully.
Steps only
Subtract 7: −8z=−2. Divide by −8: z=1/4.
Answer
z=1/4.
Question mistake-2v
Solve 2v−4=6.
Steps only
Add 4: 2v=10. Divide by 2: v=5. Dividing the 2 without dividing the −4 is invalid.
Answer
v=5.
Question mistake-5z
Solve 5z+2=3z−4.
Steps only
Subtract 3z: 2z+2=−4. Subtract 2: 2z=−6. Divide by 2: z=−3.
Answer
z=−3.
Question mistake-15w
Solve 15w−4w=26.
Steps only
Combine like terms: 11w=26. Divide by 11: w=26/11.
Answer
w=26/11.
Question mistake-3x
Solve 3x+1=−12.
Steps only
Subtract 1: 3x=−13. Divide by 3: x=−13/3.
Answer
x=−13/3.
Question mistake-4q
Solve 4(4q+2)=50.
Steps only
Expand: 16q+8=50. Subtract 8: 16q=42. Divide: q=21/8.
Answer
q=21/8.
Question mistake-distribute
Solve −2(3−4x)=14.
Steps only
Expand carefully: −6+8x=14. Add 6: 8x=20. Divide by 8: x=5/2.
Answer
x=5/2.
Question mistake-y
Solve 3(7y+4)=9+5y.
Steps only
Expand: 21y+12=9+5y. Subtract 5y and 12: 16y=−3, so y=−3/16.
Answer
y=−3/16.
Question equations-solution-minus2
Write five different equations whose solution is x=−2.
Steps only
Substitute x=−2 in each equation: both sides are equal.
Answer
Examples: x+3=1; 2x=−4; 5−x=7; 3x+1=−5; 4(x+1)=−4.
Question solve-2y
Solve 2y=60.
Steps only
Divide both sides by 2.
Answer
y=30.
Question solve-x
Solve −8=5x−3.
Steps only
Add 3: −5=5x. Divide by 5: x=−1.
Answer
x=−1.
Question solve-w
Solve −53w=−15.
Steps only
Divide both sides by −53: w=(−15)/(−53)=15/53.
Answer
w=15/53.
Question solve-z
Solve 13−z=8.
Steps only
Subtract 13: −z=−5. Multiply by −1: z=5.
Answer
z=5.
Question solve-k
Solve k+8=12−k.
Steps only
Add k: 2k+8=12. Subtract 8: 2k=4, so k=2.
Answer
k=2.
Question solve-m
Solve 7m=m−3.
Steps only
Subtract m: 6m=−3. Divide by 6: m=−1/2.
Answer
m=−1/2.
Question solve-n
Solve 3n=10+n.
Steps only
Subtract n: 2n=10. Divide by 2.
Answer
n=5.
Question three-digit
A three-digit number has hundreds digit 3 less than tens, and tens 3 less than units. The digits total 15. Find it.
Steps only
Let the hundreds digit be h. Then tens=h+3 and units=h+6. So 3h+9=15, h=2. The digits are 2, 5 and 8.
Answer
The number is 258.
Question brick
A brick weighs 1 kg more than half its own weight. Find its weight.
Steps only
Let its weight be w. Then w=w/2+1. Subtract w/2: w/2=1, so w=2.
Answer
2 kg.
Question quarter-number
One quarter of a number increased by 9 equals the number. Find it.
Steps only
Let the number be x: x/4+9=x. Subtract x/4: 9=3x/4. Multiply by 4/3: x=12.
Answer
12.
Question 4k-1
Given 4k+1=13, find (a) 8k+2, (b) 4k, (c) k, (d) 4k−1, (e) −k−2.
Steps only
Subtract 1: 4k=12, so k=3. Substitute in each expression: 8(3)+2=26; 12; 3; 12−1=11; −3−2=−5.
Answer
k=3; the requested values are 26, 12, 3, 11 and −5.
Question horse-price
One person has ₹300 and 6 horses; another has 10 horses and owes ₹100. They are equally rich. Find one horse’s price.
Steps only
Let a horse cost ₹x. 300+6x=10x−100. Add 100 and subtract 6x: 400=4x, so x=100.
Answer
₹100.
Question formula-example
Solve 2x+3=4x+5.
Steps only
Subtract 2x: 3=2x+5. Subtract 5: −2=2x. Divide by 2.
Answer
x=−1.
Question integer-blanks
Fill integers: (a) 5×□−8=37; (b) 37−(33−□)=35; (c) −3×(−11+□)=45.
Steps only
(a) 5x=45 so x=9. (b) 4+x=35 so x=31. (c) −11+x=−15 so x=−4.
Answer
(a) 9; (b) 31; (c) −4.
Question daily-wage
Ranju earns ₹750 in ₹50 and ₹100 notes, with equal numbers of each. How many notes of each?
Steps only
Let n be each type’s note count. 50n+100n=750; 150n=750; n=5.
Answer
5 of each.
Question blue-dots
There are 25 blue dots in total. Each of the 3 black blobs covers the same number of dots, and 4 dots are visible. How many dots does each blob cover? Write an equation.
Steps only
Let x be the number of dots hidden by one blob.
Three blobs hide 3x dots. Adding the 4 visible dots gives the total: 3x+4=25.
Subtract 4 from both sides: 3x=21. Divide both sides by 3: x=7.
Answer
3x+4=25, so each blob hides 7 dots.
Question taxi
A taxi costs ₹800 fixed plus ₹20 per kilometre. Total ₹2,200. Find distance.
Steps only
Let d be kilometres: 800+20d=2,200. Subtract 800: 20d=1,400. Divide by 20: d=70.
Answer
70 km.
Question sum-76
Two numbers total 76 and one is three times the other. Find them.
Steps only
Let the smaller be x. x+3x=76; 4x=76; x=19, and 3x=57.
Answer
19 and 57.
Question juice-milkshake
A juice costs ₹15 less than a milkshake. Four juices and seven milkshakes cost ₹600. Find each price.
Steps only
Let milkshake cost m; juice=m−15. 4(m−15)+7m=600, so 11m=660 and m=60. Then juice=45.
Answer
Juice ₹45; milkshake ₹60.
Question 28p-results
Given 28p−36=98, find 14p−19 and 28p−38.
Steps only
Add 36: 28p=134. Half gives 14p=67, so 14p−19=48. Also 28p−38=134−38=96.
Answer
14p−19=48; 28p−38=96.
Question triangle-angles
Triangle angles are x, y+15 and x−10 degrees (see diagram). Find x and y.
Steps only
Add the three labelled angles: x+(y+15)+(x−10)=180.
Simplify to 2x+y+5=180.
Use any extra relation shown in the original figure; without it, x and y are not unique.
Answer
The diagram’s labels and placement alone do not give another relation; using angle sum gives 2x+y+5=180. There are infinitely many pairs unless the figure supplies an additional equality/condition.
Question bakhshali
Four people receive amounts x, 2x, three times the second and four times the third. Total is 132. Find first amount.
Steps only
The amounts are x, 2x, 6x and 24x. Their total is 33x=132, so x=4.
Answer
x=2.
Question giraffe
A giraffe is 2.5 m more than half its height. Find height.
Steps only
Let height be h: h=h/2+2.5. Then h/2=2.5 and h=5.
Answer
5 m.
Question number-increased
A number increased by 36 equals ten times itself. Find the number.
Steps only
Let it be x: x+36=10x. Thus 36=9x, so x=4.
Answer
4.
Question solve-equations
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).
Steps only
Expand brackets, collect variable terms on one side, constants on the other, then divide by the coefficient. Substitute each result to check.
Answer
(a) r=0; (b) u=−4/5; (c) n=5; (d) x=−4; (e) x=0; (f) s=−1; (g) x=2; (h) x=4/3.
Question children-donkeys
Children and donkeys have 28 heads and 80 feet altogether. How many of each?
Steps only
Let d be donkeys; children=28−d. 4d+2(28−d)=80 gives 2d=24, so d=12. Then children=28−12=16.
Answer
16 children and 12 donkeys.
Question magic-trick
Explain the number trick: think x, double, add 10, halve, subtract x, then add 3.
Steps only
The operations give (2x+10)÷2−x+3 = x+5−x+3 = 8.
Answer
The result is always 8.