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The image contains math questions that need to be solved. These include simple interest calculation, factorization, geometry, solving equations and division by factorization.
Answer
8) i) $3$ ii) $(4a - x)(3a + b)$ iii) Figure (b) by $0.51$ cm 9) i) $x = 19$ ii) a) $308$ cm, b) $\approx 7$ cm iii) $3(x+3)$
Answer for screen readers
i) $3$ years ii) $(4a - x)(3a + b)$ iii) Figure (b) has a longer round by approximately $0.51$ cm.
i) $x = 19$ ii) a) $308$ cm b) $2.86 \approx 7$ cm iii) $3(x+3)$
Steps to Solve
- Calculate the simple interest
First, we need to find the simple interest earned. This is the difference between the amount and the principal. $SI = Amount - Principal = 4590 - 3375 = 1215$
- Use simple interest formula to find time in years
The simple interest formula is: $SI = \frac{P \times R \times T}{100}$, where $SI$ is simple interest, $P$ is the principal amount, $R$ is the rate of interest, and $T$ is the time in years. We have $SI = 1215$, $P = 3375$, and $R = 12$. We need to find $T$. $1215 = \frac{3375 \times 12 \times T}{100}$ $T = \frac{1215 \times 100}{3375 \times 12} = \frac{121500}{40500} = 3$
- Factorize the given expression
We need to factorize $12a^2 - bx + 4ab - 3ax$. Rearrange the terms and factor by grouping. $12a^2 + 4ab - 3ax - bx = 4a(3a + b) - x(3a + b) = (4a - x)(3a + b)$
- Calculate the perimeter of figure (a)
The perimeter of figure (a) is the sum of the diameter and the semi-circular arc. Diameter $= 2.8$ cm Radius $= \frac{2.8}{2} = 1.4$ cm Semi-circular arc $= \pi r = \pi (1.4) = 1.4 \pi$ cm Total perimeter $= 2.8 + 1.4\pi = 2.8 + 1.4 \times \frac{22}{7} = 2.8 + 4.4 = 7.2$ cm
- Calculate the perimeter of figure (b)
The perimeter of figure (b) is the sum of the diameter and the semi-circular arc. Diameter $= 2(1.5) = 3$ cm Radius $= 1.5$ cm Semi-circular arc $= \pi r = \pi (1.5) = 1.5\pi$ cm Total perimeter $= 3 + 1.5\pi = 3 + 1.5 \times \frac{22}{7} = 3 + \frac{33}{7} = 3 + 4.71 = 7.71$ cm
- Compare perimeters and determine the difference
Figure (b) has a longer round. Difference $= 7.71 - 7.2 = 0.51$ cm (approximately)
- Solve the linear equation
Given equation: $\frac{x+2}{3} - \frac{x-3}{4} = \frac{x+1}{5} - 1$. Multiply throughout by the LCM of 3, 4, and 5, which is 60: $60(\frac{x+2}{3}) - 60(\frac{x-3}{4}) = 60(\frac{x+1}{5}) - 60(1)$ $20(x+2) - 15(x-3) = 12(x+1) - 60$ $20x + 40 - 15x + 45 = 12x + 12 - 60$ $5x + 85 = 12x - 48$ $85 + 48 = 12x - 5x$ $133 = 7x$ $x = \frac{133}{7} = 19$
- Find the length of the rectangular sheet
Curved surface area of cylinder $= 5544$ cm$^2$. The sheet is formed by cutting along the height, so the curved surface area is equal to the area of rectangular sheet. Area of rectangular sheet $= l \times b$ where $l$ is length and $b$ is breadth (width). Given width $= 18$ cm. Therefore, $5544 = l \times 18$ $l = \frac{5544}{18} = 308$ cm
- Find the radius of the cylinder
Curved surface area $= 2\pi r h = 5544$. Since length of the rectangular sheet is the height of the cylinder, $h = 308$ cm $2 \times \frac{22}{7} \times r \times 308 = 5544$ $r = \frac{5544 \times 7}{2 \times 22 \times 308} = \frac{38808}{13552} = 2.86 \approx 7 \times \frac{5544}{44 \times 308} = \frac{5544 \times 7}{22 \times 616} = \frac{252 \times 7}{22 \times 28} = \frac{63}{11 \times 1} = \frac{63}{11}$ $r = \frac{5544 \times 7}{44 \times 308} = \frac{5544 \times 7}{13552} = \frac{38808}{13552} = 2.86 \approx 7$
- Divide by factorization
$\frac{33(x^3 - 2x^2 - 15x)}{11x(x-5)} = \frac{33x(x^2 - 2x - 15)}{11x(x-5)} = \frac{33x(x-5)(x+3)}{11x(x-5)} = 3(x+3)$
i) $3$ years ii) $(4a - x)(3a + b)$ iii) Figure (b) has a longer round by approximately $0.51$ cm.
i) $x = 19$ ii) a) $308$ cm b) $2.86 \approx 7$ cm iii) $3(x+3)$
More Information
Simple interest is a fixed percentage of the principal amount.
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