🧮 algebra

1. **State the problem:** We have two numbers, let's call the larger number $x$ and the smaller number $y$.

1. **State the problem:** We have 9000 liters of a 30% acid solution. We want to add water to reduce the acid concentration to 20%. 2. **Find the amount of acid in the initial solu

1. **State the problem:** Kaye is currently 24 years old and her cousin Pia is 9 years old. We need to find in how many years Kaye's age will be double Pia's age. 2. **Define varia

1. **State the problem:** Joy is 9 years older than Leo. Three years ago, the sum of their ages was 29. We need to find Joy's current age.

1. **State the problem:** Sam, Ton, and Paul together can finish a job in 3 hours. Sam alone can finish in 6 hours, and Ton alone in 8 hours. We need to find how long Paul alone wi

1. **State the problem:** Ben and Kevin start driving towards each other from two points 120 km apart. Ben drives at 45 kph and Kevin at 35 kph. We need to find the time when they

1. **State the problem:** We are given a rectangle whose width is thrice the length, and the width is equal to the side of a square.

1. **State the problem:** Roy is 13 years older than Sam. In 5 years, the sum of Roy’s age and twice Sam’s age will be 79. We need to find Sam’s current age. 2. **Define variables:

1. Start with the expression: $3\sqrt{2} + 2\sqrt{18} - 5\sqrt{50}$. 2. Simplify each square root term:

1. **State the problem:** We need to find five consecutive even integers whose sum is 470. 2. **Define variables:** Let the smallest even integer be $x$.

1. **State the problem:** Jack can carry a pail of water uphill alone in 40 minutes, Jill can do it alone in 25 minutes, and we want to find how long it takes if they help each oth

1. State the problem: Solve for $x$ in the equation $\log_2(4x - 12) = 5$.\n\n2. Recall the definition of logarithm: $\log_b(a) = c$ means $b^c = a$.\n\n3. Applying this to our equ

1. **State the problem:** Kevin can clean a room alone in 36 minutes. When Ben helps, it takes 9 minutes together. We need to find how long it takes Ben to clean alone. 2. **Define

1. **State the problem:** We need to find the smallest of four consecutive multiples of 7 whose sum is 1582. 2. **Define variables:** Let the smallest multiple be $7n$, where $n$ i

1. The problem states we have a geometric progression (GP) starting with 2, 6, 18, 54, ... and we want to find how many terms are added to get a sum of 19682. 2. Identify the first

1. **State the problem:** Find the sum of the first seven terms of an arithmetic progression (AP) where the first four terms are -7, -3, 1, and 5. 2. **Find the common difference $

1. The problem states that 𝑎 and 𝑏 are both odd numbers. 2. Recall that an odd number can be written as $2k+1$ for some integer $k$.

1. State the problem: We want to find the 20th term ($a_{20}$) of an arithmetic sequence. 2. Recall the formula for the $n$-th term in an arithmetic sequence:

1. Problem: For each pair of functions, identify one characteristic they have in common and one that distinguishes them. 2. a) Functions: $f(x) = \frac{1}{x}$ and $g(x) = x$

1. **State the problem:** We need to divide the expression $12x^7y^3 + 8xy^4 - 24x^8y^9$ by $2xty^2$. 2. **Write the division:**

1. **State the problem:** We need to find the first term $a_1$ of an arithmetic sequence given that the 19th term $a_{19} = 417$ and the 12th term $a_{12} = 319$.