Yet Things | Class 8 Mathematics Notes
By ConceptScroll Team · Published on 17 July 2026 · 5 min read
Yet Things – this guide gives you a concise, exam-ready overview of Yet Things from Class 8 Mathematics, written by ConceptScroll editors and reviewed against the latest NCERT textbook.
Some Properties of Multiplication
This section introduces the distributive property of multiplication over addition, a fundamental algebraic property that connects multiplication and addition. Starting with a practical example, consider the multiplication 23 × 27. The section explores how the product changes when one or both factors are increased by 1. For instance, increasing the second number from 27 to 28 increases the product by 23, which can be understood using the distributive property: a(b + c) = ab + ac. Here, if a = 23, b = 27, and c = 1, then 23 × (27 + 1) = 23 × 27 + 23. This property is visualized using a diagram showing the area interpretation of multiplication.
The section further generalizes this to cases where both numbers are increased by 1, leading to (a + 1)(b + 1) = ab + (a + b + 1). Similarly, if one number is increased by 1 and the other decreased by 1, the product changes by (b - a - 1). These algebraic identities hold for all integers, including negative numbers.
A general identity is presented for increases by arbitrary integers m and n: (a + m)(b + n) = ab + mb + an + mn, where the increase in product is an + bm + mn. This identity is illustrated with a visual diagram showing the multiplication of each term.
The section concludes with examples of expanding algebraic expressions using the distributive property, emphasizing the concept of like terms and simplification, and introduces the idea of algebraic identities—equations true for all values of the variables involved.
📊 Diagram: This is called the distributive property of multiplication over addition. Using the identity a(b + c) = ab + ac with a = 23, b = 27, and c = 1, we have; This identity can be visualised as follows—; See how the rules of integer multiplication allows us to handle multiple cases using a single identity!
🧪 Activity: Explore the increase in product when one or both numbers are increased or decreased by 1 by substituting various integer values for a and b, including negative integers.
🔗 Connection: This section lays the foundation for understanding algebraic expansions and identities, which are further explored in special cases of the distributive property and algebraic identities in the next sections.
Frequently asked questions
By expanding the expressions, verify that all three expressions are equivalent. If x = 8 and y = 3, find the area of the shaded region.
The three expressions given are: Vaishnavi's method: x(x + 2y) - 3xy, Aditya's method: x(x - y), and the third expression (not explicitly stated here but implied to be equivalent). Expanding Vaishnavi's expression: x(x + 2y) - 3xy = x^2 + 2xy - 3xy = x^2 - xy. Expanding Aditya's expression: x(x - y) = x^2 - xy. Both expressions are equal. Substituting x = 8 and y = 3: Area = 8^2 - 8*3 = 64 - 24 = 40. Therefore, the area of the shaded region is 40 square units.
Write an expression for the area of the dashed region in the figure below. Use more than one method to arrive at the answer. Substitute p = 6, r = 3.5, and s = 9, and calculate the area.
Method 1: Consider the larger rectangle with dimensions p and (r + s + r) = p × (2r + s). The dashed region is the area of this rectangle minus the two smaller rectangles of dimensions p × r each. Area = p(2r + s) - 2pr = p(2r + s - 2r) = p × s. Method 2: Alternatively, sum the areas of the two rectangles of dimensions p × r and the rectangle p × s, then subtract the two p × r rectangles to get the dashed area. Substituting p = 6, r = 3.5, s = 9: Area = 6 × 9 = 54 square units.
1. Compute these products using the suggested identity. (i) 462 using Identity 1A for (a + b)^2 (ii) 397 × 403 using Identity 1C for (a + b)(a – b) (iii) 912 using Identity 1B for (a – b)^2 (iv) 43 × 45 using Identity 1C for (a + b)(a – b)
(i) 462 = (400 + 60 + 2)^2 can be simplified using (a + b)^2 = a^2 + 2ab + b^2. Let a = 460, b = 2 (or better to split as 40 + 20 + 2, but here the identity is for two terms, so better to consider 40 + 22 or 400 + 62). Since 462 is a number, likely the question means 46^2. Assuming 46^2: 46^2 = (40 + 6)^2 = 40^2 + 2×40×6 + 6^2 = 1600 + 480 + 36 = 2116. (ii) 397 × 403 = (400 - 3)(400 + 3) = 400^2 - 3^2 = 160000 - 9 = 159991. (iii) 912 = (900 + 12)^2 or (a - b)^2? Assuming 91^2: 91^2 = (90 + 1)^2
2. Use either a suitable identity or the distributive property to find each of the following products. (i) (p – 1)(p + 11) (ii) (3a – 9b)(3a + 9b) (iii) –(2y + 5)(3y + 4) (iv) (6x + 5y)^2 (v) (2x – 1)^2 (vi) (7p) × (3r) × (p + 2)
(i) (p – 1)(p + 11) = p^2 + 11p – p – 11 = p^2 + 10p – 11. (ii) (3a – 9b)(3a + 9b) = (3a)^2 – (9b)^2 = 9a^2 – 81b^2. (iii) –(2y + 5)(3y + 4) = –[2y×3y + 2y×4 + 5×3y + 5×4] = –[6y^2 + 8y + 15y + 20] = –(6y^2 + 23y + 20) = –6y^2 – 23y – 20. (iv) (6x + 5y)^2 = (6x)^2 + 2×6x×5y + (5y)^2 = 36x^2 + 60xy + 25y^2. (v) (2x – 1)^2 = (2x)^2 – 2×2x×1 + 1^2 = 4x^2 – 4x + 1. (vi) (7p) × (3r) × (p + 2) = 21pr(p + 2) = 21prp + 42pr = 21p^2r + 42pr.
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