If 5 and 7 are remainders when 96 and 163 respectively are divided by a positive integer A, then A is
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Solution
91(96 - 5) and 156(163 -7) are the numbers that are perfect multiples of A. Out of the given options only 13 divides both 91 and 156 completely.
What is the greatest possible common divisor of two different positive integers which are less than 145?
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Solution
We are looking for the greatest number that is a factor of at least two different positive integers less than 145. The obvious nos. are 144 and 72, with 72 as H.C.F.
Life of a substance reduces to half at the end of one hour i.e. its quantity reduces to one half of what it was at the beginning of one hour. In how many hours, the quantity becomes less than 1% of initial quantity?
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Solution
After nth hour Qty. (% of original)
1 50
2 25
3 12.50
4 6.25
5 3.125
6 1.56
7 0.78
John can dig constantly at 15 inches per minute and Linda can dig constantly at 45 inches per minute. A certain hole can be dug by John alone in 12 hours, If hole is dug by John for half the time and both together for rest of time, how many minutes does it take to dig the hole?
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Solution
Total depth of hole = 15 x 60 x 12.
Let x⁄2 minutes be the time when John works alone and remaining ~ when both work together.
When both work together, work done is 60 inches per minute.
(x⁄5 × 15)+ +(x⁄2 × 60) = 12 × 60 × 15
∴ x⁄2 × 75 = 12 × 60 × 15 ∴ x = 288 minutes.
In a class of 120 students, boys constitute 40% of total. If X 1/3rd of boys and 4 girls drop out of class to join a camp, what % of remaining students in the class would be girls?
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Solution
Out of 120 students, 48 are boys and 72 are girls. After 16 boys and 4 girls leave, the class has 32 boys and 68 girls.
∴ Percentage of girls =\(\frac{68}{32+68}\)=68%
If a rectangle with width 49.872 inches and length 30.64 inches has an area that is 15 times the area of a certain square which of the following is closest approximation of length in inches of a side of that square?
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Solution
Approximating, length and width to 30 and 50 inches respectively, area of square
\(\frac{30\times 50}{15}=100\, inches^{2}\)
∴ Length = 10 inches.
Mark and Pat drive separately to a meeting. Mark’s average driving speed is X rd greater than Pat’s and Mark drives twice as many miles as Pat. What is ratio of number of hours Mark spends driving to the meeting to the number of hours Pat spends driving to meeting?
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Solution
Let Pat's driving speed = 3 mph
∴ Mark's driving speed = 4 mph
If Pat covers the distance of 6X miles, Mark will cover 12X miles.
∴ The ratio of time taken will be
\(\frac{12x}{4}:\frac{6x}{3}=3:2\)
A colourless cube is painted blue and then cut parallel to sides to form two rectangular solids of equal volume. What % of surface area of each of new solids is not painted blue?
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Solution
Let the side of original cube be 2.
After, cutting the cube, we get 2 rectangular solids as shown. Consider one such solid. Four sides and the top is painted blue. The bottom is not painted blue.
The required % =\(\frac{(2\times 2)}{4(2\times 1)+2(2\times 2)}=\frac{4}{16}\)= 25% .∴ Not painted blue is 25%
In a certain game of disks, each player either scores 2 points or 5 points. If N players score 2 points and M players score 5 points, and total number of points scored is 50, what is the least possible positive difference between N and M?
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Solution
The mathematical equation is 2N + 5M = 50.
To minimize the difference between the values of N and M, we have to see that N and M are as equal as possible.
We will have to use trial and error method.
\(\frac{50}{7}\) = 7 Hence M = 7.
If M = 7, then 5M = 35 and 2N = 15 which is not possible since 2N must be an even number.
If M = 8, then 5M = 40 and 2N = 10 ∴ N = 5
which gives a positive difference of 3, the least possible.
0< m < 1, which of the following has least value?
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Solution
M is a positive fraction, less than 1. Let M be 1⁄4
A=\(\frac{1}{(\frac{1}{4})^{2}}=10\: \: \: \: B=\frac{1}{\sqrt{\frac{1}{4}}}=\frac{1}{\frac{1}{2}}=2\)
C=\(\frac{1}{(\frac{1}{4})^{2}+1}+\frac{1}{\frac{1}{16}+1}=\frac{16}{17}\)
D=\(\frac{1}{\sqrt{\frac{1}{4}}+1}=\frac{1}{\sqrt{\frac{5}{4}}}=\frac{2}{2.236}\)
E=\(\frac{1}{(\frac{1}{4}+1)^{2}}=\frac{16}{25}\)