🧮 algebra

1. **State the problem:** A boy has the same number of older and younger sisters as he has older and younger brothers. Each sister has twice as many sisters as brothers. We need to

1. The problem asks us to list the numbers $\frac{1}{4}$, $\frac{4}{10}$, $\frac{41}{100}$, $0.04$, and $0.404$ from smallest to largest and find the middle number. 2. Convert all

1. The problem is to find the common divisors of the number 390. 2. First, we find the prime factorization of 390.

1. The problem is to find the Least Common Multiple (LCM) of the number 390. 2. The LCM is typically found between two or more numbers, but since only one number is given, the LCM

1. **State the problem:** Find the least common multiple (LCM) of 390 using prime factorization. 2. **Prime factorize 390:**

1. Stating the problem: Solve the equation $$(8x-7)^2 - (10x+3)(6x+6) = (2x-4)(2x+4)$$ for $x$. 2. Expand each term:

1. **State the problem:** We want to find the number of Pythagorean triples $(a,b,c)$ with positive integers such that $a^2 + b^2 = c^2$, where $a < b$ and $c = b + 2$. 2. **Substi

1. The problem asks us to list the numbers $\frac{1}{4}$, $\frac{4}{10}$, $\frac{41}{100}$, $0.04$, and $0.404$ from smallest to largest and find the middle number. 2. Convert all

1. The problem asks for the sum of all positive terms in the geometric sequence 50, -40, 32, ... 2. Identify the first term $a_1 = 50$ and the common ratio $r = \frac{-40}{50} = -0

1. **Express the following rational numbers with positive denominators:** i. $-\frac{3}{4}$ already has a positive denominator.

1. **State the problem:** Solve the simultaneous equations: $$2\log y = \log 2 + \log x$$

1. **State the problem:** We are given the equation $k^2 - xy = 2k$ with the condition $k \leq x$. We want to analyze or solve this equation under the given constraint. 2. **Rewrit

1. Let's define the variables: Let $r$ be Rachel's age. 2. Monica is a year older than Rachel, so Monica's age is $r + 1$.

1. The problem asks for the percentage decrease in traffic accidents from 1990 to the following year. 2. The initial number of accidents in 1990 was 1000.

1. **State the problem:** Given $a = -2$ and $b = 3$, find the value of $a^2 - b^3$. 2. **Calculate $a^2$:**

1. Stating the problem: We need to find the ratios of different ingredients Sara has for baking a cake. 2. Count the ingredients:

1. The problem states that a component costs 0.05 each and the manufacturer needs 2,000 components per month. 2. First, calculate the monthly cost by multiplying the cost per compo

1. **State the problem:** Simplify the expression $$(2p^2)^3(3q)^2$$. 2. **Apply the power to each factor inside the parentheses:**

1. **State the problem:** Maria and Marc both travel approximately 50 miles but at different speeds and conditions. Maria drives at 50 mph on a highway, while Marc drives at 40 mph

1. **Stating the problem:** We want to find a combination of the given numbers whose sum is exactly $1656.29$. 2. **List of numbers:** $219.25, 483.16, 163.09, 252.13, 128.67, 903.

1. **State the problem:** Simplify the expression $$3x^2 - 5x + 3 - 4 + 2x^2 - 6x$$. 2. **Group like terms:** Combine the terms with $$x^2$$, the terms with $$x$$, and the constant