If \(2^{n}=\frac{4^{5}+4^{5}+4^{5}}{3^{5}+^{5}}\times \frac{6^{5}+6^{5}+6^{5}+6^{5}}{2^{5}+ 2^{5}+ 2^{5}}\) nd n > 0, then find the value of n2
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Solution
\(2^{n}=\frac{4^{5}\times 3}{3^{5}\times 2}\times \frac{6^{5}\times 4}{2^{5}\times 3}\)
∴ \(2^{n}=\frac{2^{10}\times 3}{2\times 3^{5}}\times \frac{2^{5}\times 3^{5}\times 2^{2}}{2^{5}\times 3}\)
∴ \(2^{n}=\frac{2^{9}}{3^{4}}\times 3^{4}\times 2^{2}\)
∴ 2n = 211 ∴ n = 11 ∴ n2=121
25 men can reap a field in 20 days. After how many days 15 men can leave the work, if whole field is to be reaped in 35 days?
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Solution
25 men can reap the field in 20 days.
∴ Total work content is 500 man days.
Out of the team of 25 men, 10 men will work for all 35 days.
∴ They will contribute 350 man days.
∴ 15 men will be required to contribute only 150 man days of work content.
∴ They will be required to work only for 10 days.
In a community survey, It was 0 serve that 3⁄8 of the people are men,2⁄3 of the men have jobs and 3⁄4 of the employed men earn more than $4000 per month.2⁄3 of the people earn more than $4000 per month and 5⁄6 of the women are employed.Then the number of employed women earning more than $4000 is what fraction of total number of women?
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Solution
LCM of denominators 8, 3, 4, 6 is 48.
∴ Let there be 48 persons.
Men (3⁄8), employed(2⁄3)12
Income < $4000 (3⁄4)9
Women (5⁄8) 30, employed (5⁄6)25
2⁄3 of the people i.e. 2⁄3 of 48 = 32 people earn more than $4000.
Out of them 9 persons were men. ∴ 23 are women.
∴ Require fraction = \(\frac{23}{30}\)
Two pipes P and Q would fill a cistern in 24 and 32 minutes respectively. If both pipes are opened simultaneously, find when the first pipe must be turned off so that the cistern may be filled in exactly 16 minutes?
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Solution
We are turning off 1st pipe therefore 2nd pipe must be kept on for all 16 minutes.
2nd pipe can fill complete tank in 32 minutes.
∴ It will complete 1⁄2 tank in 1 minutes.
The remaining half will be filled by 1st pipe in 12 minutes. Since its speed is 24 minutes for full tank.
∴ 1st pipe should be turned off after 12 minutes.
A widow received (2⁄5)th of husband’s estate and each of her 3 sons received (1⁄3)rd of the balance. If the widow and one of her sons received a total of $18,000, then the amount received by one son is what fraction of the average amount received by the widow and a son?
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Solution
Let the total estate be $15 (5 × 3).
∴ Widow receives = 2⁄5 = 6
∴ Each son receives = 1⁄3 of balance $9 = $3
∴ The widow and a son together receive 6 + 3 = $9
∴ Require fraction = 3⁄9 = 1⁄3
(Value of estate is not to be considered since we are calculating only the ratio).
Paul and George trade in ready made garments. Paul correctly calculates his profit % on cost price, while George wrongly calculates it on selling price. Find the difference in their profits if both claim to make 30% profit and their selling price is $3900.
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Solution
Consider Paul' transaction -
Profit = 30%
If the Cost Price = $100, then selling price = $130
∴ If the S.P. is 3900,
then C.P. =\(\frac{3900\times 100}{130}=\$ 3000\)
∴ Profit = 3900 - 3000 = $900
Now consider George's transaction Profit
= 30% of S.P.
∴ Profit = \(\frac{30\times 3900}{100}=\$ 3000\) = $1170
∴ Difference in profits = 1170 - 900 = $270
3 liters of water is taken out from a vessel containing 10 liters of water and substituted by pure milk. The process is repeated two more times. What will be the ratio of milk and water at the end of 3rd process?
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Solution
Water Milk
Liter 10.00
-3.00 Process I
=7.00
+ 3.00 Add milk
=10.00
-2.1 - 0.9 Process II (30% of both)
=7.00
+3.00 Add milk
=10.00
-1.47 -1.53 Process III (30% of both)
=7.00
+3.00 Add milk
=10.00
Final Ratio Milk: Water = 657 : 343
= 67 : 35 Approximate quantity
Profit earned by selling the article for $1060 is 20% more than the loss incurred by selling the article for $950. What is the cost of article?
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Solution
Let loss be $x. Cost will be 950 + x
∴ 950 + x = 1060 - 1.2x
∴ 2.2x = 110 ∴x=50
∴ Cost will be 950 + 50 = 1000
Cost price is $1000.
Instead of marking up the cost price by 10%, John discounted it by 10% and prepared the price tag of a shirt. A discount of 10% is applicable to the marked price. Because of the mistake, the company earned $180 less per shirt. What is the cost price of the shirt?
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Solution
Let the cost price of shirt = $100
Then Marked Price at 10% mark up = $110
∴ Selling Price at 10% discount = 110 - 11 = $99
Market Price put by mistake = 90% of 100 = $90
Also Selling Price at 10% discount = 90 - 9 = $81
.. Difference in the two selling prices = 99 - 81 = $18
∴ When the loss is $18, Cost Price is $100
Similarly, when loss in present case is $180, the Cost Price = $1000
A man walking at a speed of 4 kmph starts from one vertex of a regular hexagonal field, walks along the diagonal and reaches the exactly opposite vertex in 3 minutes.
Find the area of the hexagonal field in m2.
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Solution
At 4 kmph he will walk 200 m in 3 minutes.
Consider the regular hexagonal field shown above. It consists of 6 equilateral triangles.
AD = 200 m. ∴ Side of triange = \(\frac{200}{3}\) = 100m.
∴ Area of the field = 6 × \(\frac{\sqrt{3}}{4}\) × (100)2 = 1500√3 m2
