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CBSE · Class 8 · Mathematics Part 2

Part 2, Chapter 6: Algebra Play

5 concepts · 40 practice questions with answers · Free to read

This lesson is also in: मराठी · हिंदी

Think of a number tricks

From daily life

At a fair a magician says: "Think of your age, add 5, double it and subtract 10. Tell me the answer and I will tell your age." You say 36. He replies 18 instantly. Algebra is his secret.

Simple definition

Call the secret number n and write each step as an expression. After simplifying, the letter n either stays (so we can work backwards from the answer) or cancels out (so the answer is fixed).

n+5n + 5×22n + 10−102n÷2n

In the figure: Add 5, double, subtract 10, halve: you return to n.

Rule 1: Write the steps — Turn each word into an operation on n. Use brackets when a later step applies to everything before it.

  • Easy: Add 3, then multiply by 4: 4(n + 3) = 4n + 12.
  • Medium: Double, then add 6: 2n + 6.
  • Exam-type: Subtract 5, then triple: 3(n − 5) = 3n − 15.

Rule 2: Simplify to find the secret — Expand brackets and combine like terms. See what is left.

  • Easy: (n + 7) × 2 − 14 = 2n: twice the start.
  • Medium: (3n + 12) ÷ 3 − 4 = n: the start itself.
  • Exam-type: (2n + 10) ÷ 2 − n = 5: always 5.

Rule 3: Work backwards — Undo the steps in reverse order, with the opposite operation.

Live figure

Think of a number tricks

In the app you can move this figure with your finger.

Open in the app

In the figure: Do the same to both sides, like undoing the trick.

  • Easy: ×4 then +9 gives 37: (37 − 9) ÷ 4 = 7.
  • Medium: ×3 then −5 gives 22: (22 + 5) ÷ 3 = 9.
  • Exam-type: Half of a number plus 6 is 15: (15 − 6) × 2 = 18.
Remember

If the letter n cancels out, the answer is the same whatever number you start with, and that is what makes a trick look like magic.

Common mistake

Forgetting brackets: "add 3 then multiply by 4" is 4(n + 3), not 4n + 3.

Shortcut

Write, simplify, look. Memory clue: "n goes in, if n cancels, magic comes out".

Spot the difference

Pointn staysn cancels
Example2n(2n + 10) ÷ 2 − n = 5
Answer depends on the start?yesno
Useguess the start from the answerpredict the answer

Practice questions

Easy
  1. 1Write as an expression: 5 more than a number tripled.
  2. 2Start with 6, add 4, multiply by 3, subtract 12. Find the result.
  3. 3A number times 5, plus 2, is 37. Find the number.
Medium
  1. 4Simplify 2(n + 5) − 10.
  2. 5Think of n, add 9, double, subtract 18, halve. What is the result in terms of n?
  3. 6A number times 3, minus 8, is 31. Find the number.
Challenging
  1. 7Think of n, multiply by 4, add 20, divide by 4, subtract n. Find the result.
  2. 8A number times 6, minus 14, then halved gives 20. Find the number.
Show answers (try them yourself first)
  1. 13n + 5
  2. 218
  3. 37
  4. 42n
  5. 5n
  6. 613
  7. 75
  8. 89

Number pyramids

From daily life

In a shop, tins are stacked so that each tin rests on two tins below. In a number pyramid each box holds the sum of the two boxes just below it.

Simple definition

For bottom row a, b, c of a 3-row pyramid the middle row is a + b, b + c and the top is a + 2b + c. For a 4-row pyramid with bottom row a, b, c, d the top is a + 3b + 3c + d.

12572342 + 3 = 5, 3 + 4 = 7, 5 + 7 = 12

In the figure: Bottom 2, 3, 4 gives 5, 7 and then 12. Check: 2 + 2 × 3 + 4 = 12.

Rule 1: Fill upward — Add each pair of neighbours to get the box above them.

Live figure

Number pyramids

In the app you can move this figure with your finger.

Open in the app

In the figure: Combine like terms the way the pyramid adds them.

  • Easy: Bottom 2, 5, 3: middle 7, 8; top 15.
  • Medium: Bottom 4, 1, 6: middle 5, 7; top 12.
  • Exam-type: Bottom 1, 9, 2: middle 10, 11; top 21.

Rule 2: Find a missing number — Use top = a + 2b + c and solve for the missing letter.

  • Easy: Top 22, bottom 3, ?, 5: 3 + 2b + 5 = 22, so b = 7.
  • Medium: Top 30, left middle 14: right middle is 30 − 14 = 16.
  • Exam-type: Top 17, bottom 2, 4, ?: 2 + 8 + c = 17, so c = 7.

Rule 3: Four rows — The two middle bottom numbers count 3 times each.

  • Easy: Bottom 1, 2, 3, 4: 1 + 6 + 9 + 4 = 20.
  • Medium: Bottom 2, 0, 1, 5: 2 + 0 + 3 + 5 = 10.
  • Exam-type: Bottom 3, 1, 1, 3: 3 + 3 + 3 + 3 = 12.
Remember

The counts 1, 2, 1 (three rows) and 1, 3, 3, 1 (four rows) tell how many times each bottom number is used in the top.

Common mistake

Adding all the bottom numbers once: 2 + 3 + 4 = 9, but the top is 12, because the middle number counts twice.

Shortcut

Three rows: 1-2-1. Four rows: 1-3-3-1. Memory clue: "121 for three, 1331 for four".

Spot the difference

Point3-row pyramid4-row pyramid
Top numbera + 2b + ca + 3b + 3c + d
Example bottom row2, 3, 41, 2, 3, 4
Top1220

Practice questions

Easy
  1. 1Bottom row 1, 4, 2 of a 3-row pyramid. Find the top.
  2. 2The middle row is 6, 9. Find the top.
  3. 3Bottom row 5, 5, 5. Find the top.
Medium
  1. 4The top is 24 and the bottom row is 3, ?, 5. Find the missing number.
  2. 5A 4-row pyramid has bottom row 1, 1, 1, 1. Find the top.
  3. 6The top is 30 and the left number of the middle row is 12. Find the right number.
Challenging
  1. 7Every bottom number of a 4-row pyramid is increased by 2. By how much does the top increase?
  2. 8A 3-row pyramid has top 40 and bottom row x, 6, x with the same x on both sides. Find x.
Show answers (try them yourself first)
  1. 111
  2. 215
  3. 320
  4. 48
  5. 58
  6. 618
  7. 716
  8. 814

Grids and calendar tricks

From daily life

A friend covers a 2 × 2 square on a calendar and says "the four dates add up to 52". You instantly say the smallest date is 9. The calendar has a pattern: one step right adds 1, one row down adds 7.

Simple definition

If the top-left date is a, the 2 × 2 square holds a, a + 1, a + 7, a + 8, with sum 4a + 16. So a = (sum − 16) ÷ 4.

aa + 1a + 7a + 84a + 16

In the figure: Sum: a + (a + 1) + (a + 7) + (a + 8) = 4a + 16.

Rule 1: Find the start from the sum — Subtract 16 and divide by 4.

  • Easy: Sum 76: (76 − 16) ÷ 4 = 15.
  • Medium: Sum 100: (100 − 16) ÷ 4 = 21.
  • Exam-type: Sum 44: (44 − 16) ÷ 4 = 7.

Rule 2: Other shapes — Write each date using a, then add. For a 3 × 3 square the sum is 9 times the middle date.

  • Easy: Three dates in a column from 5: 5 + 12 + 19 = 36 = 3a + 21.
  • Medium: Three dates in a row from 10: 10 + 11 + 12 = 33 = 3a + 3.
  • Exam-type: 3 × 3 square with middle 16: 9 × 16 = 144.

Rule 3: Grids with rules — Write what each rule says as an equation, then solve for the unknown.

  • Easy: Last = sum of first two; 6, ?, 15: ? = 15 − 6 = 9.
  • Medium: Each row adds to 15; 6, ?, 4: ? = 15 − 10 = 5.
  • Exam-type: Each row adds to 21; ?, 9, 7: ? = 21 − 16 = 5.
Remember

Every 2 × 2 calendar sum is a multiple of 4, because 4a + 16 = 4(a + 4). A sum such as 70 is impossible.

Common mistake

Using +10 for the date below. A row down on a calendar adds 7 (days in a week), not 10.

Shortcut

Right is +1, down is +7. Memory clue: "One step right, a week down".

Spot the difference

Point2 × 2 square3 × 3 square
Sum4a + 169 × middle
Find the start(sum − 16) ÷ 4middle = sum ÷ 9
Examplestart 5: 36middle 16: 144

Practice questions

Easy
  1. 1On a calendar, what date is directly below 12?
  2. 2A 2 × 2 square has top-left date 3. Find the sum.
  3. 3What date is just to the right of 20?
Medium
  1. 4A 2 × 2 square has sum 60. Find the smallest date.
  2. 5A 3 × 3 square has middle date 10. Find the sum.
  3. 6Three dates in a row start at 8. Find their sum.
Challenging
  1. 7A 2 × 2 square has sum 92. Find its largest date.
  2. 8Can a 2 × 2 calendar square have sum 70?
Show answers (try them yourself first)
  1. 119
  2. 228
  3. 321
  4. 411
  5. 590
  6. 627
  7. 727
  8. 8No (a = 13.5 is not a whole number)

The largest product

From daily life

A game gives you three digit cards, say 2, 4 and 6. You must place them as a two-digit number times a one-digit number. Which placement gives the biggest score?

Simple definition

For three different digits in □□ × □, the largest product comes when the largest digit is the multiplier and the other two form the two-digit number in decreasing order (larger digit in the tens place).

24 × 614426 × 410442 × 625246 × 29262 × 424864 × 2128

In the figure: The six products: the largest, 252, is 42 × 6.

Rule 1: Apply the rule — Pick the largest digit as the multiplier, then arrange the others big to small.

  • Easy: Digits 3, 4, 8: 43 × 8 = 344.
  • Medium: Digits 1, 5, 9: 51 × 9 = 459.
  • Exam-type: Digits 2, 6, 7: 62 × 7 = 434.

Rule 2: Compare by expanding — Split each two-digit number into tens and units. If the first parts match, compare the second parts.

  • Easy: 42 × 6 = 240 + 12 and 62 × 4 = 240 + 8, so 42 × 6 wins.
  • Medium: 43 × 8 = 320 + 24 and 83 × 4 = 320 + 12.
  • Exam-type: 31 × 8 = 240 + 8 and 81 × 3 = 240 + 3.

Rule 3: Why it works — With digits p < q < r, qp × r = 10qr + pr and rp × q = 10qr + pq. The first terms match; pr is bigger than pq because r > q.

  • Easy: Digits 2, 4, 6: difference = 2 × (6 − 4) = 4 (252 − 248).
  • Medium: Digits 1, 3, 8: 248 − 243 = 5 = 1 × (8 − 3).
  • Exam-type: Four digits 1, 2, 3, 4 in □□ × □□: 41 × 32 = 1312.
Remember

The rule works for any three different digits. For two two-digit numbers, put the two largest digits in the tens places.

Common mistake

Putting the largest digit in the tens place of the two-digit number. For 2, 4, 6 that gives 62 × 4 = 248, which is smaller than 252.

Shortcut

The biggest digit is the boss multiplier; the rest line up big to small. Memory clue: "Boss multiplies, the rest descend".

Spot the difference

PointLargest digit is multiplierLargest digit in tens place
Placement42 × 662 × 4
Product252248
Resultthe largestsmaller

Practice questions

Easy
  1. 1Digits 2, 5, 8 in □□ × □. Which digit is the multiplier for the largest product?
  2. 2Find the largest product with digits 3, 6, 7 in □□ × □.
  3. 3Which is larger, 41 × 6 or 61 × 4?
Medium
  1. 4Find the largest product with digits 1, 6, 9 in □□ × □.
  2. 5By how much is 43 × 8 larger than 83 × 4?
  3. 6Expand (10q + p) × r.
Challenging
  1. 7Use 1, 2, 3, 4 once each in □□ × □□. Find the largest product.
  2. 8Digits 1, 8, 9 in □□ × □. Find the largest product.
Show answers (try them yourself first)
  1. 18
  2. 2441
  3. 341 × 6
  4. 4549
  5. 512
  6. 610qr + pr
  7. 71312
  8. 8729

Divisibility tricks

From daily life

Asha says: "Write a two-digit number, say 83. Reverse it to 38. Subtract: 83 − 38 = 45. Divide by 9." There is never a remainder! Algebra explains why.

Simple definition

A two-digit number with tens digit a and units digit b is 10a + b. Its reverse is 10b + a. Difference: 9a − 9b = 9(a − b). Sum: 11a + 11b = 11(a + b). For a 3-digit number abc, adding its two cyclic numbers gives 111(a + b + c).

83↔38−45÷ 9510a + b − (10b + a) = 9(a − b)

In the figure: 83 − 38 = 45 and 45 ÷ 9 = 5: the difference of the digits.

Rule 1: Difference with the reverse — The difference is 9 times the difference of the digits.

  • Easy: 83 − 38 = 45 = 9 × 5.
  • Medium: 91 − 19 = 72 = 9 × 8.
  • Exam-type: 52 − 25 = 27 = 9 × 3.

Rule 2: Sum with the reverse — The sum is 11 times the sum of the digits.

  • Easy: 42 + 24 = 66 = 11 × 6.
  • Medium: 19 + 91 = 110 = 11 × 10.
  • Exam-type: 65 + 56 = 121 = 11 × 11.

Rule 3: Cyclic numbers — abc + bca + cab = 111(a + b + c) = 3 × 37 × (a + b + c), so it is divisible by 3 and 37.

  • Easy: 123 + 231 + 312 = 666 = 37 × 18.
  • Medium: 245 + 452 + 524 = 1221 = 37 × 33.
  • Exam-type: 507 + 075 + 750 = 1332 = 37 × 36.
Remember

10a + b = 9a + (a + b). The part 9a is always a multiple of 9 and of 3, so the digit sum a + b decides divisibility by 9 and by 3.

Common mistake

Trying a few examples and calling it proved. Examples can only suggest; the algebra with a and b proves it for every number.

Shortcut

Difference: 9. Sum: 11. Cyclic: 111. Memory clue: "Minus 9, Plus 11, Rotate 111".

Spot the difference

PointReverse differenceReverse sum
Expression9(a − b)11(a + b)
Always divisible by911
Example with 522777

Practice questions

Easy
  1. 1Take 83, reverse to 38, subtract and divide by 9. Find the quotient.
  2. 2Add 42 and 24, then divide by 11. Find the quotient.
  3. 3Is 738 divisible by 9?
Medium
  1. 4Add 123 and its two cyclic numbers. Divide the sum by 37.
  2. 5Write the difference of the number 10a + b and its reverse.
  3. 6Is 123 divisible by 3? By 9?
Challenging
  1. 7The number 91 and its reverse differ by 72. Write 72 as 9 times the difference of the digits.
  2. 8Why is the sum of a two-digit number and its reverse always divisible by 11?
Show answers (try them yourself first)
  1. 15
  2. 26
  3. 3Yes (7 + 3 + 8 = 18)
  4. 418
  5. 59a − 9b
  6. 6By 3 only (digit sum 6)
  7. 772 = 9 × 8
  8. 8Because it equals 11(a + b)