Practice Questions

Explain how to find the value of the expression $a - 2b$ when $a=15$ and $b=4$.

Justify why the expression $7a + 4b$ cannot be simplified further.

A pen costs ₹$x$ and a notebook costs ₹$y$. Rohan buys 5 pens and 3 notebooks. His friend Karan buys 2 pens and 4 notebooks. Formulate an expression for the total amount spent by both of them and simplify it.

Name the components of the term $-9y$.

An equilateral triangle has a side length of $s$ cm. Write an algebraic expression for its perimeter and then calculate the perimeter if $s = 8$ cm.

Identify the terms in the algebraic expression $5a - 3b + 2c - 7$.

A student claims that the expressions $5(k-2)$ and $5k-2$ are equivalent. Critique this statement by providing a numerical example to prove or disprove it.

A student simplified the expression $(8p - 5q) - (3p - 7q)$ as $5p - 12q$. Critique the student's method, identify the error, and provide the correct simplification with a clear explanation.

Calculate the value of the expression $7p - 12$ when $p = 5$.

Explain the difference between the algebraic expressions $x^2$ and $2x$. Use $x=5$ to support your explanation.

Define the term 'letter-number' as used in algebra.

Describe in words what the following algebraic expressions represent: (a) $m-1$ (b) $5n+3$

Describe a real-world scenario that can be represented by the algebraic expression $100 + 5x$.

Summarize the steps to write an algebraic expression for the statement: "10 less than the product of a number $x$ and a number $y$".

Explain the concept of simplifying an algebraic expression. Use the expression $3p + 5q + 2p - q$ to illustrate the process.

Sunita's current age is $y$ years. Her father is 5 years older than three times her age. Her brother is 4 years younger than her. Write expressions for her father's age and her brother's age. Then, calculate the sum of their three ages.

A rectangular field has a length of $(3x+4)$ meters and a breadth of $(2x-1)$ meters. Analyze the dimensions to find an expression for the perimeter of the field. Calculate the perimeter if $x=10$ meters.

Design a matchstick pattern where the number of matchsticks required for the $n$-th step is given by the formula $4n + 1$. Draw the first three steps of your designed pattern and verify that the formula holds.

An equilateral triangle has all three sides of equal length. (a) Describe how to find the perimeter of an equilateral triangle. (b) If the length of one side is denoted by the letter-number $s$, explain how to write a formula for the perimeter, $P$. (c) Explain what makes this formula an algebraic expression.

A pattern is formed using matchsticks to create a chain of hexagons. The first step (one hexagon) requires 6 matchsticks. The second step (two hexagons) requires 11 matchsticks. The third step (three hexagons) requires 16 matchsticks.

a) Analyze the pattern to find an algebraic expression for the number of matchsticks required to form $n$ hexagons. b) Calculate the number of matchsticks needed to form 15 hexagons.

Two friends, Rohan and Priya, are saving money. Rohan starts with ₹200 and saves ₹50 each week. Priya starts with ₹350 and saves ₹25 each week. Formulate an expression for the total savings of each after w weeks. Evaluate which person will have more savings after 8 weeks and by how much.

Create an algebraic expression containing at least three terms that simplifies to $2x + 5y$.

Examine the expression $9x + 4y - 5x - y$. Combine the like terms to simplify it.

Solve for the value of the expression $a(a+b) - b^2$ if $a=4$ and $b=3$.

Formulate a real-world problem for which the expression $150 - 5x$ represents the remaining amount of money.

List the mathematical operations present in the expression $4(x+5)$.

Identify which of the following pairs are 'like terms' and explain why: (a) $3x$ and $3y$ (b) $5p$ and $-2p$ (c) $4k^2$ and $6k$

Recall the standard way to write the expression for 'the product of 7 and a number $k$'.

A taxi service charges a flat fee of ₹50 and an additional ₹15 per kilometer. Formulate an algebraic expression for the total cost of a journey of $k$ kilometers.

Simplify the expression $5(x+2y) - 3(x-y)$ and then calculate its value for $x=3$ and $y=2$.

Subtract the expression $4a - 7b + 10$ from $15a - 9b + 2$.

A rectangular park has a length that is 5 meters less than twice its breadth. Formulate an expression for its perimeter in terms of its breadth, $b$. If the perimeter is 74 meters, justify the steps to find the park's dimensions.

A taxi service 'QuickCab' charges a flat fee of ₹50 and an additional ₹12 per kilometer. Another service 'GoFast' charges a flat fee of ₹30 and ₹15 per kilometer. Formulate an expression for the cost of travelling d kilometers for each service. Determine for what distance d the cost for both services would be equal.

Ajay, Bimal, and Chetan have marbles. Bimal has 10 fewer marbles than twice the number of marbles Ajay has. Chetan has 5 more marbles than Bimal. Let $x$ be the number of marbles Ajay has.

a) Write algebraic expressions for the number of marbles Bimal and Chetan have. b) Formulate a simplified expression for the total number of marbles they have altogether. c) If Ajay has 20 marbles, calculate the total number of marbles.

Explain why the expression $4(a+b)$ is not the same as $4a+b$.

Describe how a formula is a special type of algebraic expression. Use the formula for the area of a triangle, $A = \frac{1}{2}bh$, to explain your points.

Two expressions are given: A = $x + (x+7)$ and B = $2(x+3.5)$. Evaluate if these expressions are always equal. Justify your answer.

Design a word problem involving the cost of two types of items (e.g., pens and notebooks) and a discount. The final cost should be represented by the simplified algebraic expression $8x + 5y - 50$, where x is the number of pens and y is the number of notebooks. Describe the original prices and the discount in your problem statement and justify how your problem leads to the given expression.

Consider any $3 \times 3$ square of dates from a calendar. Propose that the sum of all nine dates is always nine times the center date. Justify this proposal using algebraic expressions.

A family's monthly budget for groceries is ₹$g$ and for utilities is ₹$u$. In April, they spent exactly their budgeted amount. In May, their grocery expenses increased by 10% and their utility expenses decreased by ₹500. In June, their grocery expenses were ₹1000 less than in May, and their utility expenses were 5% more than in April.

a) Write an expression for the total expenses in May. b) Write a simplified expression for the total expenses in June. c) Calculate the total expenses for all three months combined.

Compare the expressions $3(n+4)$ and $3n+4$. Are they equal? Justify your answer by calculating their values for $n=1$ and $n=5$.

An artist makes a pattern of squares from tiles. Step 1 is a single square. In Step 2, squares are added to form a larger $2 \times 2$ square. In Step 3, squares are added to form a $3 \times 3$ square. a) Formulate an expression for the total number of small squares in Step n. b) Formulate an expression for the perimeter of the entire shape in Step n, assuming each small square has a side length of 1 unit. c) Justify that the number of new squares added to get from Step n-1 to Step n is $2n - 1$.

A school is planning a trip. The cost is ₹1200 per student. For every 10 students that sign up, the organizing committee gets a group discount of ₹500. A student formulates the total cost for s students as $1200s - 500 \times \frac{s}{10}$. a) Critique this formula. Identify a scenario where it fails or gives a nonsensical answer. b) Propose a more accurate way to describe the cost calculation and justify why your proposed method is better.

Prove algebraically that the sum of any three consecutive odd integers is always a multiple of 3. Justify each step of your proof.

What should be added to $5x^2 - 3xy + 7y^2$ to get $8x^2 + 2xy - 4y^2$?