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Questions and Answers

In the matchstick pattern for the letter 'N', if 2 matchsticks are needed to form the vertical lines, and one matchstick for the diagonal, what is the algebraic expression for the number of matchsticks required to make 'x' number of 'N's?

  • $x + 3$
  • $3 + x$
  • $3x$ (correct)
  • $2x + 1$

Which rule accurately describes the progression in the number pattern: 250, 130, 70, 40, 25?

  • Divide by 2 and subtract 5.
  • Divide the previous number by 2 and add 5. (correct)
  • Multiply by 2 and subtract 370.
  • Subtract 120 and add 10.

What does the equation $\frac{m}{7} - 12 = 23$ represent?

  • One-seventh of a number is 12 less than 23. (correct)
  • Seven times a number is 12 less than 23.
  • Seven times a number is 12 more than 23.
  • One-seventh of a number is 12 more than 23.

If $\frac{100a}{2} = 500$, what is the value of 'a'?

<p>10 (A)</p> Signup and view all the answers

If Reema has $12a$ flowers and Ria has 6 more flowers than Reema, how many flowers does Ria have?

<p>$12a + 6$ (B)</p> Signup and view all the answers

Rahul spends $y$ daily and saves $z$ per week. What is his income for three weeks?

<p>$21y + 3z$ (D)</p> Signup and view all the answers

What is the 15th odd number in the sequence of odd numbers derived using the formula $2n - 1$?

<p>29 (B)</p> Signup and view all the answers

Akash has $t$ toffees. His brother Amit has 6 toffees less than Akash. How many toffees do they have together?

<p>$2t - 6$ (C)</p> Signup and view all the answers

Sandeep was $p$ years old 18 years ago. If $p = 13$, how old is Sandeep now?

<p>31 (B)</p> Signup and view all the answers

Consider a pattern of pentagons formed using matchsticks. If $n$ represents the number of pentagons, which expression accurately calculates the total number of matchsticks required?

<p>$5n + 1$ (D)</p> Signup and view all the answers

Using the formula for the $n^{th}$ odd number ($2n - 1$), determine which odd number in the sequence is equal to 45.

<p>The $23^{rd}$ odd number (A)</p> Signup and view all the answers

A pattern consists of repeating a unit made of matchsticks. The smallest unit has 6 uncommon matchsticks and 2 common matchsticks. What is the general formula for the total number of matchsticks required for $n$ units?

<p>$6n + 2$ (C)</p> Signup and view all the answers

A sequence of figures is created using matchsticks. The first figure requires 7 matchsticks, and each subsequent figure requires 5 additional matchsticks. Determine the number of matchsticks needed to create the 10th figure in the sequence.

<p>52 (C)</p> Signup and view all the answers

Which of the following expressions represents the total number of pencils Anwar has, if he puts them into x boxes holding 15 pencils each, with 3 pencils left over?

<p>$15x + 3$ (A)</p> Signup and view all the answers

If Jyoti's current age is represented by $x$, which expression represents Mira's age 4 years ago, given that Mira is 3 years older than Jyoti?

<p>$x - 1$ (C)</p> Signup and view all the answers

Ali's age is represented by $x$. Lucky is 4 times as old as Ali. What expression represents Lucky's age 6 years from now?

<p>$4x + 6$ (D)</p> Signup and view all the answers

In a school, $x$ students are divided into 3 equal groups for a cricket tournament. If 4 cricketers are added to each group, which expression represents the number of cricketers in each group now?

<p>$\frac{x}{3} + 4$ (D)</p> Signup and view all the answers

Pooja's present age is $y$ years. Which expression correctly represents her mother's age, if her mother is 5 years older than twice Pooja's age?

<p>$2y + 5$ (C)</p> Signup and view all the answers

Sakshi is $x$ years old. Her uncle is $(3x - 4)$ years old. Which of the following statements accurately reflects this relationship?

<p>Sakshi's uncle's age is three times Sakshi's age decreased by 4 years. (C)</p> Signup and view all the answers

The cost of a shampoo bottle is $c$. The cost of a detergent pack is $(2c + 3)$. What does this algebraic expression tell us in ordinary language?

<p>The detergent pack costs $3 more than twice the cost of the shampoo bottle. (A)</p> Signup and view all the answers

Flashcards

Formula for nth odd number

The nth odd number can be found by multiplying n by 2 and subtracting 1.

Matchsticks for one inverted C

One inverted 'C' requires three matchsticks.

Total Matchsticks for 'n' inverted C

For 'n' inverted Cs, you need 2n + 1 matchsticks.

Matchstick Pattern Generalization

Count the distinct (uncommon) matchsticks in one unit, then add the shared (common) ones.

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Matchsticks in repeated pentagons

Five uncommon matchsticks in the smallest repeated unit and 1 common matchstick.

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Repeated pentagons

This pattern is generated by adding 'n' number of pentagons.

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Repeated pattern

A repeated pattern of 'C's is formed by inverted 'C' units placed adjacent to each other.

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General formula.

The uncommon matchsticks in the smallest repeated unit added to the number of units and the number of common matchsticks in the smallest repeated unit.

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Solution of an equation

A value that makes the equation true.

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Trial and error method

Finding the correct value by guessing and checking.

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Algebraic equation

A mathematical statement showing equality using variables.

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Balancing equations

Performing the same operation on both sides to keep the equation equal.

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General rule

A general formula showing the relationship between numbers using variables.

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Variable

An unknown quantity represented by a letter.

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Equation

A statement where two expressions are equal.

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Translating word problems

Convert words into algebraic expressions or equations.

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What is an equation?

An equation is a statement showing the equality between two expressions or numbers.

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What is LHS?

The expression on the left side of the equals sign (=) in an equation.

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What is RHS?

The expression on the right side of the equals sign (=) in an equation.

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Expressions like 3y > 9. Equations?

These are not equations because these contain inequality.

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Jaya's age after 10 years?

If Jaya is x years old now, Jaya's age after 10 years.

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Jyoti's age 4 years ago?

If Jyoti is x years old now, Jyoti's age 4 years ago.

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If Ali is x, Lucky's age 6 years from now

Lucky's age is 4x + 6

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Expression for cricketers in each group.

x/3 + 4

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Study Notes

Algebraic Expressions Basics

  • Algebra uses letters and symbols to represent unknown numbers.
  • These letters, called variables, can take on different values.
  • Arithmetic is the branch of mathematics that deals with numbers having fixed values and basic operations.

Algebraic Expressions

  • Algebraic expressions combine variables and constants connected by mathematical operations.
  • Examples of algebraic expressions include 2x + 9.
  • a + 5 means '5 added to a,' '5 more than a,' or 'a increased by 5.'
  • a - 5 means '5 subtracted from a,' '5 taken away from a,' or 'a decreased by 5.'
  • 5a represents 5 multiplied by a, or 5 times a.
  • a/5 represents a divided by 5 or one-fifth of a.
  • To evaluate an algebraic expression, substitute a given value for the variable.

Commutative and Distributive Properties

  • The commutative property allows changing the order of numbers in multiplication (e.g., 2 x 5 = 5 x 2, or x * y = y * x).
  • The distributive property can be written as x * (y + z) = x * y + x * z.

Even and Odd Numbers

  • Numbers in the times table of 2 are even numbers, expressible as 2n.
  • Odd numbers are expressible as 2n - 1.

Matchstick Patterns

  • Patterns can be generalized to find the number of matchsticks needed to make 'n' inverted C's: 2n + 1
  • The total number of matchsticks in a repeated unit pattern can be generalized.
  • You use: uncommon matchsticks + (number of units) + common matchsticks.

Equations

  • An equation is a statement of equality between two expressions.
  • The left-hand side (LHS) must equal the right-hand side (RHS).
  • If LHS does not equal RHS, it is not an equation.

Solving Equations

  • A 'solution' or 'root' satisfies an equation.
  • The trial and error method involves substituting values to find the solution.
  • Balancing an equation means performing the same operation on both sides.

Practical Applications

  • Algebra can represent real-world scenarios.
  • For example, the cost of b packs of biscuits at ₹10 each is 10b.

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