🧮 algebra

1. **Problem Statement:** Ramu borrowed 1000 at 10% per annum simple interest for 3 years, but after the first year, the interest rate was revised to 12% per annum. We need to find

1. **State the problem:** A photograph measures 7 \(\frac{1}{2}\) cm (which is 7.5 cm) by 5 cm. It is enlarged so that the longer side becomes 24 cm. We need to find the length of

1. **State the problem:** We have a room with dimensions 30 ft long, 25 ft wide, and 14 ft high. There are 42 balloons inside the room, and we want to find how many cubic feet of s

1. **State the problem:** We need to find how many passengers, each weighing 50.5 kg, can fit in an elevator with a maximum load capacity of 605 kg. 2. **Formula used:** To find th

1. **Problem statement:** Determine whether the quadratic equation $x^2 + 4x + 4 = 0$ has distinct real roots, equal real roots, or no real roots. 2. **Formula used:** For a quadra

1. The problem is to understand the difference between $np$ and $n^p$ in mathematical expressions. 2. The expression $np$ means multiplication of $n$ and $p$, i.e., $n \times p$.

1. **Problem Statement:** Find the roots of the quadratic equation $$x^2 - 5x + 6 = 0$$. 2. **Formula Used:** The quadratic formula for roots of $$ax^2 + bx + c = 0$$ is $$x = \fra

1. The problem is to convert the quadratic equations from standard form to vertex form. 2. The standard form of a quadratic equation is $$y = ax^2 + bx + c$$.

1. The problem is to understand and verify the logarithmic identity: $\log_a b = \frac{1}{\log_b a}$. 2. Recall the change of base formula for logarithms: $\log_a b = \frac{\log_c

1. **State the problem:** Convert the point-slope form equation $$y + 3 = 5(x + 2)$$ into slope-intercept form $$y = mx + b$$. 2. **Recall the formula:** Point-slope form is $$y -

1. **State the problem:** Convert the point-slope form equation $$y - 4 = -(x + 2)$$ to slope-intercept form $$y = mx + b$$. 2. **Recall the point-slope form:** $$y - y_1 = m(x - x

1. **Stating the problem:** We need to verify or solve the equation $\log_b a \cdot \log_c b \cdot \log_a c = 1$. 2. **Recall the change of base formula:** For any positive numbers

1. **State the problem:** Convert the point-slope form equation $$y + 2 = 5(x - 3)$$ to slope-intercept form $$y = mx + b$$. 2. **Recall the formula:** The point-slope form is $$y

1. **State the problem:** Convert the point-slope form equation $$y + 3 = 4(x - 5)$$ into slope-intercept form $$y = mx + b$$ where $$m$$ is the slope and $$b$$ is the y-intercept.

1. **State the problem:** Convert the point-slope form equation $$y + 5 = -2(x - 1)$$ to slope-intercept form $$y = mx + b$$. 2. **Recall the formula:** The point-slope form is $$y

1. **Stating the problem:** We want to verify the identity $\log_a b \times \log_c a = \log_c b$. 2. **Recall the change of base formula:** For any positive numbers $x, y$ and base

1. **State the problem:** We start with the line $y = x$ and transform it into $y = \frac{x}{4} + 2$. We need to find the slope and describe how the line is shifted. 2. **Recall th

1. **Problem Statement:** Expand the expression $$\left(4x^3 + \frac{1}{2}x\right)^8$$ using the binomial theorem. 2. **Formula:** The binomial theorem states:

1. **Stating the problem:** Three friends share a sum of money in the ratio 2:3:5. One friend gets 1200. We need to find how much more one friend gets compared to another. 2. **Und

1. **Stating the problem:** We are given pairs of numbers connected by arrows: 90 \rightarrow 60, 30 \rightarrow 30, 18 \rightarrow 22, 12 \rightarrow 0, and a horizontal line with

1. **State the problem:** We are given the ratio of the present ages of two people, $a$ and $b$, as $3:4$. Five years ago, the ratio of their ages was $2:3$. We need to find their