🧮 algebra

1. **State the problem:** We want to find how many integers between 100 and 999 have an odd ones digit, an even tens digit, and an even hundreds digit. 2. **Analyze the digits:**

1. The problem states: Let $$198 = 2 + 1a + 1b + 1c$$. We need to find the value of $$a + b + c$$. 2. Rewrite the equation clearly: $$198 = 2 + a + b + c$$.

1. Stating the problem: Simplify the expression $$6(x+5)^2 - 2(x-4)^2$$. 2. Expand each squared term:

1. **State the problem:** We need to find two positive integers whose sum is 60 and whose least common multiple (LCM) is 273. 2. **Recall the relationship between two numbers, thei

1. **Stating the problem:** We have a row of 3 squares made using 10 toothpicks. We need to find how many toothpicks are needed to make a row of 11 squares. 2. **Understanding the

1. The problem is to simplify the expression $$(2x+3)^2 - (9-4x)^2$$. 2. Recognize this as a difference of squares: $$a^2 - b^2 = (a-b)(a+b)$$ where $$a = 2x+3$$ and $$b = 9-4x$$.

1. Stating the problem: Simplify the expression $$(x-6)^2 - (x-6)(x+6)$$. 2. Expand each term:

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.