Mathematics: Class 7 (Part 1) — Chapter 4: EXPRESSIONS USING LETTER-NUMBERS

Shabnam is 3 years older than Aftab. If Aftab's age is 10 years, Shabnam's age is 13 years. If Aftab's age is 18 years, Shabnam's age will be 21 years.

• To find Shabnam's age given Aftab's age, add 3 to Aftab's age.
• This can be written as: Shabnam's age = Aftab's age + 3.
• Mathematical relations are often represented in a shorthand form using symbols (letters or short phrases).
• Letter-numbers: Letters like $a$ (for Aftab's age) and $s$ (for Shabnam's age) used to represent numbers.
• Algebraic expressions: Mathematical expressions containing letter-numbers, such as $a + 3$.
• The expression to find Shabnam's age is $s = a + 3$.
• If $a = 23$ (Aftab's age), then $s = 23 + 3 = 26$ years.

• Aftab is 3 years younger than Shabnam.
• Expression: Aftab's age = Shabnam's age $– 3$.
• Using letters: $a = s – 3$.
• If Shabnam's age ($s$) is 20, Aftab's age ($a$) is $20 – 3 = 17$ years.

Parthiv makes L-shaped patterns with matchsticks. Each L uses 2 matchsticks.

• Number of matchsticks for 5 Ls: $5 \times 2 = 10$.
• Number of matchsticks for 7 Ls: $7 \times 2 = 14$.
• Number of matchsticks for 45 Ls: $45 \times 2 = 90$.
• The relation: Number of matchsticks = $2 \times$ Number of L's.
• Using $n$ for the number of L's, the algebraic expression is $2 \times n$ or $2n$.

Price of a coconut: ₹35. Price of 1 kg jaggery: ₹60.

• Cost for 10 coconuts and 5 kg jaggery: $10 \times \text{₹}35 + 5 \times \text{₹}60 = \text{₹}350 + \text{₹}300 = \text{₹}650$.
• Cost for 8 coconuts and 9 kg jaggery: $8 \times \text{₹}35 + 9 \times \text{₹}60 = \text{₹}280 + \text{₹}540 = \text{₹}820$.
• To find the total amount for $c$ coconuts and $j$ kg jaggery:
  • Cost of coconuts: $c \times 35$.
  • Cost of jaggery: $j \times 60$.
  • Total amount: $c \times 35 + j \times 60$.
• If $c = 7$ and $j = 4$, total amount: $7 \times 35 + 4 \times 60 = 245 + 240 = \text{₹}485$.

• The perimeter of a square is 4 times the length of its side.
• Expression: $4 \times q$, where $q$ is the sidelength.
• If sidelength $q = 7$ cm, perimeter is $4 \times 7 = 28$ cm.
• Letter-numbers and algebraic expressions allow expressing general mathematical relations concisely, often called formulas.

• 1. Write formulas for the perimeter of:
  • (a) Triangle with all sides equal: $3a$ (where $a$ is side length).
  • (b) Regular pentagon: $5a$ (where $a$ is side length).
  • (c) Regular hexagon: $6a$ (where $a$ is side length).
• 2. Combined length of pipes: Munirathna has a 20 m pipe and joins another pipe of length $k$ meters. Expression: $(20 + k)$ meters.
• 3. Total amount Krithika has:

  Table showing number of notes and total amount:

  • 3 ₹100, 5 ₹20, 6 ₹5: $3 \times 100 + 5 \times 20 + 6 \times 5 = 300 + 100 + 30 = 430$.
  • 6 ₹100, 4 ₹20, 3 ₹5: $6 \times 100 + 4 \times 20 + 3 \times 5 = 600 + 80 + 15 = 695$.
  • 8 ₹100, 4 ₹20, $z$ ₹5: $8 \times 100 + 4 \times 20 + z \times 5 = 800 + 80 + 5z = 880 + 5z$.
  • $x$ ₹100, $y$ ₹20, $z$ ₹5: $x \times 100 + y \times 20 + z \times 5 = 100x + 20y + 5z$.
• 4. Time taken for flour mill: 10 seconds to start, 8 seconds per kg of grain. For $y$ kg of grain, expression: $10 + 8 \times y$ or $10 + 8y$.
• 5. Write algebraic expressions: (Let $d$ be the number)
  • (a) 5 more than a number: $d + 5$.
  • (b) 4 less than a number: $d - 4$.
  • (c) 2 less than 13 times a number: $13d - 2$.
  • (d) 13 less than 2 times a number: $2d - 13$.

Identify and correct mistakes in simplifying expressions:

• 1. If $a = -4$, then $10 - a = 6$.
  • Mistake: $10 - (-4) = 10 + 4 = 14$.
  • Correct value: $14$.
• 2. If $d = 6$, then $3d = 36$.
  • Mistake: $3 \times 6 = 18$.
  • Correct value: $18$.
• 3. If $s = 7$, then $3s - 2 = 15$.
  • Mistake: $3 \times 7 - 2 = 21 - 2 = 19$.
  • Correct value: $19$.
• 4. If $r = 8$, then $2r + 1 = 29$.
  • Mistake: $2 \times 8 + 1 = 16 + 1 = 17$.
  • Correct value: $17$.
• 5. If $j = 5$, then $2j = 10$.
  • Correct: $2 \times 5 = 10$.
• 6. If $m = -6$, then $3(m + 1) = 19$.
  • Mistake: $3(-6 + 1) = 3(-5) = -15$.
  • Correct value: $-15$.
• 7. If $f = 3, g = 1$, then $2f - 2g = 2$.
  • Mistake: $2 \times 3 - 2 \times 1 = 6 - 2 = 4$.
  • Correct value: $4$.
• 8. If $t = 4, b = 3$, then $2t + b = 24$.
  • Mistake: $2 \times 4 + 3 = 8 + 3 = 11$.
  • Correct value: $11$.
• 9. If $h = 5, n = 6$, then $h - (3 - n) = 4$.
  • Mistake: $5 - (3 - 6) = 5 - (-3) = 5 + 3 = 8$.
  • Correct value: $8$.

• Perimeter = length + breadth + length + breadth.
• Using letter-numbers $l$ for length and $b$ for breadth: $p = l + b + l + b$.
• Rearranging terms: $p = l + l + b + b$.
• Simplifying: $p = 2l + 2b$.
• The expression $2l + 2b$ is the simplified form of $l + b + l + b$.
• Example: If $l = 3, b = 4$, then $l + b + l + b = 3 + 4 + 3 + 4 = 14$, and $2l + 2b = 2 \times 3 + 2 \times 4 = 6 + 8 = 14$.

Price per pencil is $c$, price per eraser is $d$.

• Money earned from pencils:
  • Day 1: $5c$
  • Day 2: $3c$
  • Day 3: $10c$
• Total money from pencils: $5c + 3c + 10c = (5 + 3 + 10)c = 18c$.
• If $c = 4$, total earned from pencils: $18 \times 50 = 900$.
• Money earned from erasers:
  • Day 1: $4d$
  • Day 2: $6d$
  • Day 3: $1d$
• Total money from erasers: $4d + 6d + 1d = (4 + 6 + 1)d = 11d$.
• Total money earned from pencils and erasers: $18c + 11d$.
• This expression cannot be simplified further as it contains different letter-numbers.
• Like terms: Terms involving the same letter-numbers (e.g., $5c, 3c, 10c$). They can be added and simplified.
• Unlike terms: Terms with different letter-numbers (e.g., $18c, 11d$). They cannot be combined.

A big rectangle is split into two smaller rectangles with side lengths $v$ and $4$, and $v$ and $3$.

• Area of smaller rectangles: $4v$ and $3v$.
• Area of bigger rectangle (sum of smaller areas): $4v + 3v$.
• Area of bigger rectangle (direct calculation): $v \times (4 + 3) = 7v$.
• Thus, $4v + 3v = 7v$, demonstrating the distributive property.
• For a rectangle AEFD with sides $n$ and $(12 - 4)$: Area = $n \times (12 - 4) = 8n$.
• Alternatively, Area = $12n - 4n = 8n$.

Rent: Chair ₹40, Table ₹75. Return: Chair ₹6, Table ₹10.

• Total amount paid at beginning for $x$ chairs and $y$ tables: $40x + 75y$.
• Total amount returned for $x$ chairs and $y$ tables: $6x + 10y$.
• Net amount paid = $(40x + 75y) - (6x + 10y)$.
• Simplifying the expression:
  • $(40x + 75y) - 6x - 10y$
  • Group like terms: $(40x - 6x) + (75y - 10y)$
  • Result: $(40 - 6)x + (75 - 10)y = 34x + 65y$.

Scores in three rounds: $7p - 3q$, $8p - 4q$, and $6p - 2q$. ($p$ = correct score, $q$ = penalty).

• If $p = 4, q = 1$:
• Round 1 score: $7 \times 4 - 3 \times 1 = 28 - 3 = 25$.
• Round 2 score: $8 \times 4 - 4 \times 1 = 32 - 4 = 28$.
• Round 3 score: $6 \times 4 - 2 \times 1 = 24 - 2 = 22$.
• Total score after three rounds: $(7p - 3q) + (8p - 4q) + (6p - 2q)$.
• Removing brackets and grouping like terms:
• $7p + 8p + 6p - 3q - 4q - 2q$
• $(7 + 8 + 6)p - (3 + 4 + 2)q$
• $21p - 9q$.

• Using distributive property: $4x + 4y - y$.
• Combine like terms: $4x + (4 - 1)y = 4x + 3y$.

• $5u$ means 5 times $u$.
• $5 + u$ means 5 more than $u$.
• These expressions are generally not equal.
• Example values:
  • If $u = 2$: $5u = 10$, $5 + u = 7$.
  • If $u = 5$: $5u = 25$, $5 + u = 10$.
  • If $u = 8$: $5u = 40$, $5 + u = 13$.
  • If $u = 11$: $5u = 55$, $5 + u = 16$.

• $10y - 3$ means 3 less than 10 times $y$.
• $10(y - 3)$ means 10 times (3 less than $y$).
• These expressions are generally not equal.
• Example values:
  • If $y = 0$: $10y - 3 = -3$, $10(y - 3) = -30$.
  • If $y = 2$: $10y - 3 = 17$, $10(y - 3) = -10$.
  • If $y = 7$: $10y - 3 = 67$, $10(y - 3) = 40$.
  • If $y = 10$: $10y - 3 = 97$, $10(y - 3) = 70$.

• 1. Add the numbers in each picture:
  • Picture 1: $5y, -6, x, x, 2, 5y$. Sum: $10y + 2x - 4$.
  • Picture 2: $2p, 3q, -2, 3q, 2p, 3, 3q, 2p, 3q, 2p$. Sum: $8p + 12q + 2$.
  • Picture 3: $-5g, 5k, 5k, -5g, 5k, 5k, 5k, 5k, 5k, 5k, 5k, 5k, -5g, 5k, 5k, -5g$. Sum: $-20g + 60k$.
• 2. Simplify each of the following expressions:
  • (a) $p+p+p+p, p+p+p+q, p+q+p-q$
    • $4p$
    • $3p+q$
    • $2p$
  • (b) $p-q+p-q, p+q-p+q$
    • $2p-2q$
    • $2q$
  • (c) $p+q-(p+q), p-q-p-q$
    • $0$
    • $-2q$
  • (d) $2d-d-d-d, 2d-d-d-c$
    • $-d$
    • $-c$
  • (e) $2d-d-(d-c), 2d-(d-d)-c$
    • $c$
    • $2d-c$
  • (f) $2d-d-c-c$
    • $d-2c$
• Mind the Mistake, Mend the Mistake (Simplification Errors):

  Correct Simplest Forms:

  • 1. $3a + 2b$: Correct.
  • 2. $3b - 2b - b$: $0$. Correct.
  • 3. $6(p + 2)$: $6p + 12$. (Mistake: $6p + 8$).
  • 4. $(4x + 3y) - (3x + 4y)$: $x - y$. (Mistake: $x + 5$).
  • 5. $5 - (2 - 6z)$: $3 + 6z$. (Mistake: $3 - 6z$).
  • 6. $2 + (x + 3)$: $x + 5$. (Mistake: $2x - 6$).
  • 7. $2y + (3y - 6)$: $5y - 6$. (Mistake: $-y + 6$).
  • 8. $7p - p + 5q - 2q$: $6p + 3q$. (Mistake: $7p + 3q$).
  • 9. $5(2w + 3x + 4w)$: $30w + 15x$. (Mistake: $10w + 15x + 20w$).
  • 10. $3j + 6k + 9h + 12$: Correct.
  • 11. $4(2r + 3s + 5)$: $8r + 12s + 20$. (Mistake: $-20 - 8r - 12s$).
• Question: Is there any relation between the number of terms and the number of letter-numbers in simplified expressions? (No direct relation, depends on the expression).

A number machine takes two inputs (top numbers of 'Y') and produces a result (bottom number).

• Inputs: $(5, 2) \implies 8$.
• Inputs: $(8, 1) \implies 15$.
• Inputs: $(9, 11) \implies 7$.
• Inputs: $(10, 6) \implies 14$.
• Inputs: $(10, 4) \implies 16$.
• The formula is
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