This problem involves profit and loss calculations. We need to determine the cost price of the rubber-bands first, based on the given selling price and loss percentage. Then, we can calculate the selling price required to achieve a 20% gain and find out how many rubber-bands must be sold for a rupee at that price.
A man sells 6 rubber-bands for a rupee. So, the selling price (SP) of 1 rubber-band is:
\[ \text{SP of 1 rubber-band} = \frac{1}{6} \text{ Rupee} \]The man loses 40% by selling at this price. This means the selling price is 60% of the cost price.
\[ \text{SP} = \text{CP} \times (1 - \text{Loss Percentage}) \]\[ \frac{1}{6} = \text{CP} \times (1 - 0.40) \]\[ \frac{1}{6} = \text{CP} \times 0.60 \]\[ \text{CP} = \frac{1}{6 \times 0.60} = \frac{1}{3.6} \text{ Rupee} \]To gain 20%, the new selling price (SP') should be 120% of the cost price.
\[ \text{SP'} = \text{CP} \times (1 + \text{Gain Percentage}) \]\[ \text{SP'} = \frac{1}{3.6} \times (1 + 0.20) \]\[ \text{SP'} = \frac{1}{3.6} \times 1.20 \]\[ \text{SP'} = \frac{1.20}{3.6} = \frac{12}{36} = \frac{1}{3} \text{ Rupee} \]If the new selling price of 1 rubber-band is \(\frac{1}{3}\) Rupee, then for 1 Rupee, the man must sell:
\[ \text{Number of rubber-bands} = \frac{1 \text{ Rupee}}{\text{SP' of 1 rubber-band}} \]\[ \text{Number of rubber-bands} = \frac{1}{\frac{1}{3}} = 3 \]Therefore, to gain 20%, he must sell 3 rubber-bands for a rupee.
C) 3 — As calculated, selling 3 rubber-bands for a rupee will result in a 20% gain.
The problem states that a number, when increased by 100%, results in 2780. We need to find the original number. Increasing a number by 100% means adding the number itself to the original number. This is equivalent to doubling the number.
A) 1390 — As calculated, when 1390 is increased by 100% (which is 1390 itself), it becomes \(1390 + 1390 = 2780\).
We are given that the product of two consecutive natural numbers is 210. We need to find the greater of these two numbers. We can solve this by setting up an algebraic equation or by testing the options.
Alternatively, we can quickly estimate the square root of 210. \(\sqrt{210}\) is between \(\sqrt{196} = 14\) and \(\sqrt{225} = 15\). Since the numbers are consecutive, they must be 14 and 15. Their product is \(14 \times 15 = 210\). The greater number is 15.
Correct Option: C) 15
Given that 1490.59 = x, 1490.94 = y and xz = y6, then the value of z is close to:
The problem involves solving for an unknown variable \(z\) using given exponential equations and a relationship between them. We will use the properties of exponents, specifically that if \(a^m = a^n\), then \(m=n\), and the property \((a^m)^n = a^{mn}\).
Given:
\[ x = 149^{0.59} \]\[ y = 149^{0.94} \]The relationship is \(x^z = y^6\).
Substitute the expressions for \(x\) and \(y\) into this equation:
\[ (149^{0.59})^z = (149^{0.94})^6 \]The rule is \((a^m)^n = a^{mn}\). Applying this rule to both sides of the equation:
\[ 149^{(0.59 \times z)} = 149^{(0.94 \times 6)} \]Since the bases are the same (149) on both sides of the equation, the exponents must be equal:
\[ 0.59 \times z = 0.94 \times 6 \]Now the equation is:
\[ 0.59 \times z = 5.64 \]\[ z = \frac{5.64}{0.59} \]Rounding to two decimal places, \(z \approx 9.56\).
C) \(9.56\) is the value closest to the calculated value of \(z\).
To find the average of the first 151 even numbers, we need to identify the series of these numbers and then use the formula for the average of an arithmetic progression, or simply the average of the first and last term for a sequence with a constant difference.
A) 152 is the correct average of the first 151 even numbers.
We need to find the total number of registered voters. We are given information about the percentage of voters who cast their vote, invalid votes, the winning candidate's share of valid votes, and the margin of victory. We will work backward from the margin of victory to find the total registered voters.
Let the total number of registered voters be \(V\).
80% of the registered voters cast their vote. So, the number of votes polled is:
\[ \text{Votes Polled} = 0.80 \times V \]5% of the votes polled were found invalid. This means 95% of the votes polled were valid. So, the number of valid votes is:
\[ \text{Valid Votes} = 0.95 \times (\text{Votes Polled}) \]\[ \text{Valid Votes} = 0.95 \times (0.80 \times V) \]\[ \text{Valid Votes} = 0.76 \times V \]The winning candidate got 75% of the valid votes. So, the losing candidate got \(100\% - 75\% = 25\%\) of the valid votes.
Winning candidate's votes: \(0.75 \times (\text{Valid Votes})\)
Losing candidate's votes: \(0.25 \times (\text{Valid Votes})\)
The winning candidate won by a margin of 1805 votes. This margin is the difference between the votes received by the winning candidate and the losing candidate.
\[ \text{Margin} = (\text{Winning candidate's votes}) - (\text{Losing candidate's votes}) \]\[ 1805 = 0.75 \times (\text{Valid Votes}) - 0.25 \times (\text{Valid Votes}) \]\[ 1805 = (0.75 - 0.25) \times (\text{Valid Votes}) \]\[ 1805 = 0.50 \times (\text{Valid Votes}) \]Now, we can find the total number of valid votes:
\[ \text{Valid Votes} = \frac{1805}{0.50} \]\[ \text{Valid Votes} = 3610 \]We know that \(\text{Valid Votes} = 0.76 \times V\). We can now find \(V\):
\[ 3610 = 0.76 \times V \]\[ V = \frac{3610}{0.76} \]\[ V = 4750 \]A) 4750 is the total number of registered voters, calculated by working backward from the margin of victory through valid votes, polled votes, and finally to the total registered voters.
Pipe A can fill a tank in 16 hours, pipe B can fill the same tank in 25 hours and pipe C can fill the same tank in 12 hours. The time taken by them to fill the same tank if they operate together is:
5 \( \frac{85}{223} \) hours
3 \( \frac{85}{223} \) hours
1 \( \frac{85}{223} \) hours
9 \( \frac{85}{223} \) hours
To find the time taken by multiple pipes to fill a tank together, we first determine the individual filling rates of each pipe. The rate is the reciprocal of the time taken to fill the tank. Then, we sum these individual rates to get the combined filling rate. Finally, the total time taken to fill the tank together is the reciprocal of the combined rate.
Calculate the filling rate of Pipe A:
Pipe A fills the tank in 16 hours. So, in 1 hour, Pipe A fills \( \frac{1}{16} \) of the tank.
Calculate the filling rate of Pipe B:
Pipe B fills the tank in 25 hours. So, in 1 hour, Pipe B fills \( \frac{1}{25} \) of the tank.
Calculate the filling rate of Pipe C:
Pipe C fills the tank in 12 hours. So, in 1 hour, Pipe C fills \( \frac{1}{12} \) of the tank.
Calculate the combined filling rate of all three pipes:
When all three pipes operate together, their combined rate is the sum of their individual rates:
Find a common denominator for the fractions:
The least common multiple (LCM) of 16, 25, and 12 is 1200.
Add the fractions:
\[ \text{Combined Rate} = \frac{1 \times 75}{16 \times 75} + \frac{1 \times 48}{25 \times 48} + \frac{1 \times 100}{12 \times 100} \]\[ \text{Combined Rate} = \frac{75}{1200} + \frac{48}{1200} + \frac{100}{1200} \]\[ \text{Combined Rate} = \frac{75 + 48 + 100}{1200} \]\[ \text{Combined Rate} = \frac{223}{1200} \text{ (of the tank per hour)} \]Calculate the total time taken to fill the tank:
The time taken is the reciprocal of the combined rate:
Convert the improper fraction to a mixed fraction:
Divide 1200 by 223:
Correct Option: A) \( 5 \frac{85}{223} \) hours is the correct time taken for all three pipes to fill the tank together.
To find the mean proportional of two numbers, we use the formula that states the mean proportional (let's call it \(x\)) of two numbers \(a\) and \(b\) is given by the square root of their product, i.e., \(x = \sqrt{a \times b}\).
B) 20 is the correct answer because, as calculated, the mean proportional of 25 and 16 is 20.
To find the trade discount percentage, we first need to calculate the actual trade discount amount. The trade discount is the difference between the list price and the price at which the retailer bought the item (cost price for the retailer). Then, we express this discount as a percentage of the list price.
The trade discount is the reduction from the list price that the retailer receives. In this context, the price at which the retailer bought the fan is effectively the price after the trade discount has been applied to the list price.
Trade Discount Amount = List Price - Retailer's Cost Price
\[ \text{Trade Discount Amount} = \text{₹}3,600 - \text{₹}2,800 = \text{₹}800 \]The trade discount percentage is calculated with respect to the List Price.
Trade Discount Percentage = \[\left( \frac{\text{Trade Discount Amount}}{\text{List Price}} \right) \times 100\]
\[ \text{Trade Discount Percentage} = \left( \frac{800}{3600} \right) \times 100 \]\[ \text{Trade Discount Percentage} = \left( \frac{8}{36} \right) \times 100 \]\[ \text{Trade Discount Percentage} = \left( \frac{2}{9} \right) \times 100 \]\[ \text{Trade Discount Percentage} = 0.2222... \times 100 \]\[ \text{Trade Discount Percentage} = 22.22...\% \]Rounding 22.22...% to one decimal place gives 22.2%.
D) 22.2% is the correct trade discount percentage when rounded to one decimal place.
This problem involves relating distance, speed, and time. We will set up equations based on the given information for Salaj and Kevin's travel times and speeds, and then solve these equations to find Salaj's speed.
Let the speed of Salaj be \(S_S\) km/hr and the speed of Kevin be \(S_K\) km/hr.
The distance to be covered is 182 km.
The time taken by Salaj is \(T_S = \frac{182}{S_S}\) hours.
The time taken by Kevin is \(T_K = \frac{182}{S_K}\) hours.
According to the first condition, Salaj takes 4 hours more than Kevin:
\[ T_S = T_K + 4 \]\[ \frac{182}{S_S} = \frac{182}{S_K} + 4 \quad \text{(Equation 1)} \]According to the second condition, if Salaj doubles his speed, his new speed would be \(2S_S\). Let the new time taken by Salaj be \(T'_S\).
\[ T'_S = \frac{182}{2S_S} \]In this case, Salaj would take 9 hours less than Kevin. Note that Kevin's speed and time remain the same.
\[ T'_S = T_K - 9 \]\[ \frac{182}{2S_S} = \frac{182}{S_K} - 9 \quad \text{(Equation 2)} \]Now we have a system of two equations. Let's subtract Equation 2 from Equation 1:
\[ \left(\frac{182}{S_S} - \frac{182}{2S_S}\right) = \left(\frac{182}{S_K} + 4\right) - \left(\frac{182}{S_K} - 9\right) \]\[ \frac{182}{S_S} - \frac{91}{S_S} = \frac{182}{S_K} + 4 - \frac{182}{S_K} + 9 \]\[ \frac{182 - 91}{S_S} = 13 \]\[ \frac{91}{S_S} = 13 \]Solve for \(S_S\):
\[ S_S = \frac{91}{13} \]\[ S_S = 7 \]So, Salaj's speed is 7 km/hr.
A) 7 km/hr is the correct speed for Salaj, as calculated from the given conditions.