Class 7 Mathematics Expressions Using Letter Numbers Study Notes

Fundamentals of Letter Numbers and Algebraic Expressions

• Letter numbers (also referred to as variables) are letters used in mathematics to represent numbers or quantities that can vary or take on different values.

• An algebraic expression is a mathematical expression containing letter numbers (variables), constants (fixed numbers), and operational signs ($+$, $-$, $\times$, $\text{or} \times$).

• Age Relationship Example (Shabnam and Aftab):

  • Shabnam is $3$ years older than Aftab.

  • Let Shabnam's age be represented by the letter number $s$ (or $S$).

  • Let Aftab's age be represented by the letter number $a$ (or $A$).

  • The algebraic equation representing this relationship is:     $s = a + 3$

  • Case 1: When Aftab is $10$ years old ($a = 10$):     $s = 10 + 3 = 13$     Shabnam's age is $13$ years.

  • Case 2: When Aftab is $18$ years old ($a = 18$):     $s = 18 + 3 = 21$     Shabnam's age is $21$ years.

  • In this expression, $a$ represents a number that varies ($10$, $18$, etc.), and $s$ takes the corresponding evaluated value ($13$, $21$, etc.).

Real-World Applications and General Expression Building

• Ketki's Coconut and Jaggery Laddus:

  • Ketki prepares and supplies coconut-jaggery (nariyal and gud) laddus.

  • Price of $1$ coconut = $\text{Rupees } 35$

  • Price of $1\text{ kg}$ jaggery = $\text{Rupees } 60$

  • Specific Purchase Calculation:

  • Cost for $10$ coconuts: $10 \times 35 = \text{Rupees } 350$

  • Cost for $5\text{ kg}$ jaggery: $5 \times 60 = \text{Rupees } 300$

  • Total amount paid: $350 + 300 = \text{Rupees } 650$

  • General Algebraic Expression:

  • Let the number of coconuts bought be $c$

  • Let the quantity of jaggery bought in kg be $j$

  • Total cost of coconuts = $35 \times c = 35c$

  • Total cost of jaggery = $60 \times j = 60j$

  • Total amount to be paid = $35c + 60j$

  • Verification: Substituting $c = 10$ and $j = 5$ into $35c + 60j$ yields $35(10) + 60(5) = 350 + 300 = 650$.

• Munirata's Garden Watering Pipe:

  • Initial pipe length owned = $20\text{ meters}$

  • Munirata joins an additional pipe of length $k\text{ meters}$ to extend the overall length.

  • Combined total length expression = $(20 + k)\text{ meters}$

  • Addition is used because the total length requires adding the joined section to the original pipe length.

• Kritika's Currency Notes:

  • Kritika holds currency notes of denominations $\text{Rupees } 100$, $\text{Rupees } 20$, and $\text{Rupees } 5$.

  • Combination 1: $3$ notes of $\text{Rupees } 100$, $5$ notes of $\text{Rupees } 20$, and $6$ notes of $\text{Rupees } 5$:     $\text{Total Amount} = 3 \times 100 + 5 \times 20 + 6 \times 5 = 300 + 100 + 30 = \text{Rupees } 430$

  • Combination 2: $6$ notes of $\text{Rupees } 100$, $4$ notes of $\text{Rupees } 20$, and $3$ notes of $\text{Rupees } 5$:     $\text{Total Amount} = 6 \times 100 + 4 \times 20 + 3 \times 5 = 600 + 80 + 15 = \text{Rupees } 695$

  • Combination 3: $8$ notes of $\text{Rupees } 100$, $4$ notes of $\text{Rupees } 20$, and $z$ notes of $\text{Rupees } 5$:     $\text{Total Amount} = 8 \times 100 + 4 \times 20 + 5 \times z = 800 + 80 + 5z = 880 + 5z$

  • General Combination: $x$ notes of $\text{Rupees } 100$, $y$ notes of $\text{Rupees } 20$, and $z$ notes of $\text{Rupees } 5$:     $\text{Total Amount} = 100x + 20y + 5z$

Translating Verbal Statements to Algebraic Expressions

• Algebraic expressions require converting word phrases into mathematical operations:

  • Phrases indicating addition:

  • "$5$ more than a number": $x + 5$ or $5 + x$

  • Demonstrates the Commutative Property of Addition: $a + b = b + a$

  • Phrases indicating subtraction:

  • "$4$ less than a number": $x - 4$

  • Multi-operation phrases:

  • "$2$ less than $13$ times a number": $13x - 2$

  • "$13$ less than $2$ times a number": $2x - 13$

• Venkatesh's Flour Mill Application:

  • Roller mill startup time delay = $10\text{ seconds}$

  • Grinding rate per kilogram of grain = $8\text{ seconds}$

  • Time required to grind $y\text{ kg}$ of grain = $8 \times y = 8y\text{ seconds}$

  • Total time expression to complete grinding $y\text{ kg}$ of grain = $10 + 8y\text{ seconds}$

  • Operational misconception caution: Expression $(10 + 8)y = 18y$ is mathematically incorrect because the startup time of $10\text{ seconds}$ is a constant initial delay, not a rate multiplied by the quantity of grain $y$.

Calendar Grid Expressions and Common Evaluation Pitfalls

• $2 \times 3$ Calendar Grid Expression:

  • Selecting a $2 \times 3$ grid of consecutive dates on a calendar:

  • If the bottom middle cell date is denoted by $w$:

  • Cell directly to the left of $w$ = $w - 1$

  • Cell directly to the right of $w$ = $w + 1$

  • Cell directly above $w$ (1 week prior) = $w - 7$

  • Cell top-left (directly above $w - 1$) = $(w - 7) - 1 = w - 8$

  • Cell top-right (directly above $w + 1$) = $(w - 7) + 1 = w - 6$

  • Summary of grid positions:

  • Top Row: $w - 8$, $w - 7$, $w - 6$

  • Bottom Row: $w - 1$, $w$, $w + 1$

• Common Errors to Avoid during Algebraic Evaluation:

  • Incorrect Substitution with Negative Numbers:

  • Evaluating $10 - a$ when $a = -4$:

  • Incorrect logic: Dropping the negative sign and calculating $10 - 4 = 6$

  • Correct evaluation: $10 - (-4) = 10 + 4 = 14$

  • Improper Concatenation vs Multiplication:

  • Evaluating $3d$ when $d = 6$:

  • Incorrect logic: Placing digits side-by-side to write $36$

  • Correct evaluation: $3 \times d = 3 \times 6 = 18$

  • Evaluating $3s - 2$ when $s = 7$:

  • Incorrect logic: Concatenating to get $37 - 2 = 35$

  • Correct evaluation: $3 \times 7 - 2 = 21 - 2 = 19$

  • Evaluating Terms with Multiple Variables:

  • Evaluating $2fg - 2g$ when $f = 3$ and $g = 1$:

  • Correct evaluation: $2 \times 3 \times 1 - 2 \times 1 = 6 - 2 = 4$

  • Incorrect Expansion of Brackets with Negative Signs:

  • Evaluating $h - (3 - n)$ when $h = 5$ and $n = 6$:

  • Correct expansion: Distributing the negative sign changes the signs inside the bracket:       $h - 3 + n = 5 - 3 + 6 = 8$

  • Alternative correct evaluation order: $5 - (3 - 6) = 5 - (-3) = 5 + 3 = 8$

Simplification of Algebraic Expressions, Like Terms, and Unlike Terms

• Pencil and Eraser Store Sales Example:

  • Price per pencil = $c$

  • Price per eraser = $d$

  • Sales over three days:

  • Day 1: $5$ pencils ($5c$), $4$ erasers ($4d$)

  • Day 2: $3$ pencils ($3c$), $6$ erasers ($6d$)

  • Day 3: $10$ pencils ($10c$), $1$ eraser ($1d$)

  • Total money earned from pencils = $5c + 3c + 10c = 18c$

  • Total money earned from erasers = $4d + 6d + 1d = 11d$

  • Total shopkeeper revenue = $18c + 11d$

• Definitions and Classification of Terms:

  • Like Terms: Terms containing the exact same letter numbers (variables) raised to the exact same exponents (powers).

  • Examples: $5c$ and $6c$ are like terms; $7y$ and $19y$ are like terms.

  • Coefficients (numerical factors) are ignored when determining if terms are like terms.

  • Unlike Terms: Terms containing different letter numbers or the same letter numbers raised to different exponents.

  • Examples: $6c^2$ and $4c$ are unlike terms because the exponents on $c$ differ ($2$ versus $1$); $7y$ and $8x$ are unlike terms because the variables differ ($y$ versus $x$).

  • Fundamental Rule of Algebraic Operations: Only like terms can be added or subtracted together by operating on their numerical coefficients.

Furniture Rental and Operation Properties

• Furniture Rental Deposit and Refund Problem:

  • Chair deposit per piece = $\text{Rupees } 40$

  • Chair refund upon return = $\text{Rupees } 6$

  • Net rental price per chair = $40 - 6 = \text{Rupees } 34$

  • Table deposit per piece = $\text{Rupees } 75$

  • Table refund upon return = $\text{Rupees } 10$

  • Net rental price per table = $75 - 10 = \text{Rupees } 65$

  • For renting $x$ chairs and $y$ tables:

  • Total deposit paid upfront = $40x + 75y$

  • Total payback refunded = $6x + 10y$

  • Net payment formula = $(40x + 75y) - (6x + 10y) = 40x + 75y - 6x - 10y = 34x + 65y$

  • Equivalent algebraic expression expansion: $40x + 75y + (-6x - 10y)$

• Application of the Distributive Property:

  • Simplifying $4(x + y) - y$:     $4(x + y) - y = 4x + 4y - y = 4x + 3y$

• Non-Equivalence Demonstration ($5u$ versus $5 + u$):

  • Multiplication ($5u$) and addition ($5 + u$) yield entirely different values for identical variable inputs:

  • If $u = 11$: $5u = 5(11) = 55$, whereas $5 + u = 5 + 11 = 16$

  • If $u = 8$: $5u = 5(8) = 40$, whereas $5 + u = 5 + 8 = 13$

  • If $u = 2$: $5u = 5(2) = 10$, whereas $5 + u = 5 + 2 = 7$

  • If $u = 5$: $5u = 5(5) = 25$, whereas $5 + u = 5 + 5 = 10$

  • Conclusion: $5u \neq 5 + u$

Expressions with Operations and Pattern Recognition

• Simplifying Sequences of Variables:

  • $p + p + p + p + p + p + p + q = 7p + q$

  • $p + p - q - q = 2p - 2q = 2(p - q)$

  • $(p + q) - (p + q) = p + q - p - q = 0$

  • $2d - d - c - c = d - 2c$

• Rules of Signs for Integers in Terms:

  • Same Signs: Add absolute values and preserve the common sign ($-1c - 1c = -2c$).

  • Opposite Signs: Subtract the smaller absolute value from the larger absolute value and retain the sign of the number with the larger absolute value ($-14c + 16c = 2c$).

• Numeric Input-Output Pattern Expressions:

  • Pattern 1:

  • Data pairs: $(5, 2) \rightarrow 8$; $(8, 1) \rightarrow 15$; $(9, 11) \rightarrow 7$; $(6, 4) \rightarrow 8$

  • Mathematical rule: Multiply the 1st number by $2$, then subtract the 2nd number.

  • Expression for inputs $x$ (1st number) and $y$ (2nd number): $2x - y$

  • Pattern 2:

  • Data pairs: $(5, 2) \rightarrow 5$; $(8, 1) \rightarrow 7$; $(9, 11) \rightarrow 18$; $(10, 10) \rightarrow 18$

  • Mathematical rule: Add the 1st number and 2nd number together, then subtract $2$

  • Expression for inputs $a$ (1st number) and $b$ (2nd number): $a + b - 2$

Periodic Design Patterns and Calendar Mathematical Structures

• Saree Border Periodic Designs:

  • A repeating cycle consists of three distinct designs: Design A, Design B, and Design C (Period $k = 3$):

  • Position 1: A, Position 2: B, Position 3: C

  • Position 4: A, Position 5: B, Position 6: C

  • Position 7: A, Position 8: B, Position 9: C

  • Mathematical rules based on division remainders when position number $n$ is divided by $3$:

  • Design C: Occurs at positions where $n$ is divisible by $3$ (remainder = $0$, form $3k$).

    • Example: Position $27$ ($27 \times \frac{1}{3} = 9$, remainder $0$) $\rightarrow$ Design C.

    • Example: Position $99$ ($99 \times \frac{1}{3} = 33$, remainder $0$) $\rightarrow$ Design C.

  • Design A: Occurs at positions where $n$ leaves remainder $1$ when divided by $3$ (form $3k + 1$).

    • Example: Position $148$ ($148 = 3 \times 49 + 1$) $\rightarrow$ Design A.

  • Design B: Occurs at positions where $n$ leaves remainder $2$ when divided by $3$ (form $3k + 2$).

    • Example: Position $8$ ($8 = 3 \times 2 + 2$) $\rightarrow$ Design B.

• Calendar Diagonal Properties ($2 \times 2$ Blocks):

  • Any $2 \times 2$ block selected from a calendar page takes the algebraic form:

  • Top-left cell = $a$

  • Top-right cell = $a + 1$

  • Bottom-left cell = $a + 7$

  • Bottom-right cell = $a + 8$

  • Diagonal Addition:

  • Diagonal 1 sum: $a + (a + 8) = 2a + 8$

  • Diagonal 2 sum: $(a + 1) + (a + 7) = 2a + 8$

  • Conclusion: Diagonal sums of any $2 \times 2$ calendar section are identically equal regardless of the starting date $a$

• Calendar Cross/Plus Pattern Centered at Cell $a$:

  • Selecting a 5-cell cross array centered at date $a$:

  • Center cell = $a$

  • Left cell = $a - 1$

  • Right cell = $a + 1$

  • Top cell (1 week prior) = $a - 7$

  • Bottom cell (1 week later) = $a + 7$

  • Total Sum of Cross Cells:     $\text{Sum} = (a - 7) + (a - 1) + a + (a + 1) + (a + 7) = 5a$

  • Conclusion: The sum of numbers in a calendar cross configuration is always exactly $5$ times the center number $a$.

  • Example: Center date = $15$

  • Total cross sum = $5 \times 15 = 75$

• Matchstick Connected Patterns:

  • Building connected triangular/polygonal shapes using matchsticks:

  • Shape 1 ($1$ unit): $3$ matchsticks

  • Shape 2 ($2$ connected units): $3 + 2 = 5$ matchsticks

  • Shape 3 ($3$ connected units): $5 + 2 = 7$ matchsticks

  • General expression for $y$ connected shapes: $3 + 2(y - 1) = 2y + 1$

Extended Word Problems, Addition, and Subtraction Rules

• Jowar Roti and Pulao Ordering Problem:

  • Price per plate of Jowar Roti = $\text{Rupees } 30$

  • Price per plate of Pulao = $\text{Rupees } 20$

  • Total cost for ordering $x$ plates of Jowar Roti and $y$ plates of Pulao = $30x + 20y$

• Snail Climbing a Deep Well:

  • Daytime climb distance = $u\text{ cm}$

  • Nighttime slip distance = $d\text{ cm}$

  • Net daily progress (1 day + 1 night) = $(u - d)\text{ cm}$

  • Total progress after $10$ days and $10$ nights = $10u - 10d$ or $10(u - d)\text{ cm}$

• Local Train Travel Time (Yahanpur to Wahanpur):

  • The train line contains $3$ intermediate stations spaced at equal distances.

  • Total journey segments between consecutive stations = $4$

  • Time taken to travel between consecutive stations = $t\text{ minutes}$

  • Total movement time = $4t\text{ minutes}$

  • Dwell/stop time at each intermediate station = $2\text{ minutes}$

  • Total dwell time at $3$ intermediate stations = $3 \times 2 = 6\text{ minutes}$

  • Expression for total journey duration = $4t + 6\text{ minutes}$

  • Evaluation for $t = 4\text{ minutes}$:     $\text{Total time} = 4(4) + 6 = 16 + 6 = 22\text{ minutes}$

• Algebraic Polynomial Addition:

  • Problem: Add $(8d - 14c + 9)$ and $(16c - 11 - 9d)$

  • Rearranging like terms:     $(8d - 9d) + (-14c + 16c) + (9 - 11)$

  • Simplifying each component:

  • $8d - 9d = -1d = -d$

  • $-14c + 16c = 2c$

  • $9 - 11 = -2$

  • Simplified result: $-d + 2c - 2$

• Algebraic Polynomial Subtraction Rule:

  • Critical Rule: Statements formatted as "Subtract $A$ from $B$" MUST be written as $B - A$.

  • Problem: Subtract $(9a - 6b + 14)$ from $(6a - 9b - 18)$

  • Algebraic Setup: $(6a - 9b - 18) - (9a - 6b + 14)$

  • Distributing the negative sign through the second expression:     $6a - 9b - 18 - 9a + 6b - 14$

  • Grouping like terms:

  • $a$ terms: $6a - 9a = -3a$

  • $b$ terms: $-9b + 6b = -3b$

  • Constant terms: $-18 - 14 = -32$

  • Simplified result: $-3a - 3b - 32$