a) By selling an article at shs 22,500/= a shopkeeper makes loss of 10%. At what price must the shopkeeper…

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a) By selling an article at shs 22,500/= a shopkeeper makes loss of 10%. At what price must the shopkeeper sell the article in order to get a profit of 10%? b) Find the sum of the first four terms of

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3. MATHEMATICS

a) By selling an article at shs 22,500/= a shopkeeper makes loss of 10%. At what price must the shopkeeper sell the article in order to get a profit of 10%?

Calculate the Cost Price (CP).
A loss of 10% means the selling price is 90% of the cost price.
$0.90 \times \text{CP} = 22,500$ $\text{CP} = \frac{22,500}{0.90}$ $\text{CP} = 25,000 \text{ shs}$
Calculate the selling price for a 10% profit.
To make a profit of 10%, the selling price must be 110% of the cost price.
\text{Selling Price (for 10% profit)} = 1.10 \times \text{CP} \text{Selling Price (for 10% profit)} = 1.10 \times 25,000 \text{Selling Price (for 10% profit)} = 27,500 \text{ shs}
The shopkeeper must sell the article at $\boxed{27,500 \text{ shs}}$ to get a profit of 10%.

b) Find the sum of the first four terms of a geometric progression which has a first term of 1 and a common ratio of $\frac{1}{4}$ .
Identify the given values.
First term ( $a$ ) = 1
Common ratio ( $r$ ) = $\frac{1}{4}$
Number of terms ( $n$ ) = 4
Use the formula for the sum of the first $n$ terms of a geometric progression.
Since $|r| < 1$ , the formula is $S_n = \frac{a(1-r^n)}{1-r}$ .
Substitute the values and calculate.
$S_4 = \frac{1 \left(1 - \left(\frac{1}{4}\right)^4\right)}{1 - \frac{1}{4}}$ $S_4 = \frac{1 - \frac{1}{256}}{\frac{3}{4}}$ $S_4 = \frac{\frac{256 - 1}{256}}{\frac{3}{4}}$ $S_4 = \frac{\frac{255}{256}}{\frac{3}{4}}$ $S_4 = \frac{255}{256} \times \frac{4}{3}$ $S_4 = \frac{255 \times 4}{256 \times 3}$ $S_4 = \frac{85 \times 1}{64 \times 1} \quad (\text{since } 255/3 = 85 \text{ and } 256/4 = 64)$ $S_4 = \frac{85}{64}$
The sum of the first four terms is $\boxed{\frac{85}{64}}$ .

4. MATHEMATICS

a) The sum of the first six terms of an A.P is 72 and the second term is seven times the fifth term. Find the first term and the common difference.
Formulate equations from the given information.
Let the first term be $a$ and the common difference be $d$ .
The sum of the first $n$ terms of an A.P. is $S_n = \frac{n}{2}(2a + (n-1)d)$ .
For $n=6$ , $S_6 = 72$ :
$72 = \frac{6}{2}(2a + (6-1)d)$ $72 = 3(2a + 5d)$ $24 = 2a + 5d \quad \text{(Equation 1)}$
The $n$ -th term of an A.P. is $a_n = a + (n-1)d$ .
Second term ( $a_2$ ) = $a + d$ .
Fifth term ($a

A) By selling an article at shs 22,500/= a shopkeeper makes loss of 10%. At what price must the shopkeeper sell the article in order to get a profit of 10%? b) Find the sum of the first four terms of

Step-by-step solution

Calculate the Cost Price (CP).

Calculate the selling price for a 10% profit.

Identify the given values.

Use the formula for the sum of the first $n$ terms of a geometric progression.

Substitute the values and calculate.

Formulate equations from the given information.

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