# Quant Capsules by Shashank Prabhu - Set 10

• How many natural numbers among first 2050 natural numbers cannot be written as sum of two or more consecutive natural numbers?

Trial and error works best if you do not know the concept... all odd numbers can be written as a sum of consecutive integers except 1 and even numbers that are multiples of an odd number can also be written as sum of consecutive integers. The only ones that remain are the ones in the form of 2^n. So from n = 0 to 11 we have 12 numbers.

Sameep cheats both while buying and selling goods. While he buying goods from the shopkeeper he uses a weighing machine that shows 1000 gm when the actual quantity is 1100 gm and while selling he uses a machine that shows 1100 gm but actual is 1000 gm.if he sells the goods at cp determine the profit percentage in whole.
20%
21%
11%
None of these.

Assuming it is 1 Rs per gram, while buying he will pay 1000 Rs for 1100 gm. So the quantity with him is 1100 gm.
While selling the machine shows 1100 gm when actual is 1000 gm. But he has 1100 gm which he wants to sell.
What will be shown when actual is 1100 gm?
Cross multiply to get 1210. So he will get paid 1210 Rs, for selling that quantity.
1210 - 1000 = 210/1000 = 21

The other approach is to use a + b + ab/100 = 10 + 10 + 1 = 21% profit

A watch is being shown in a museum which has a very peculiar property. It gains as much in the day as it loses during night between 8 PM to 8 AM. In a week how many times will the clock show the correct time?

Let's say the week starts at 8 am in the morning... the clock will show the right time at the start and the gap would start to increase till the time the actual time is 8 pm. Then, it will start falling back and will show the correct time at 8 am the next morning. Over a week, this will happen 8 times.

Three men and 5 women together can finish a job in 3 days. Working on the same job 3 women take 5 days more than the time required by 2 men. What is the ratio of efficiency of a man to a woman?
(1) 2 : 1
(2) 3 : 2
(3) 5 : 2
(4) 4 : 1
(5) None of these

3m+5w=work done by 3 men and 5 women in 1 day
Total work will be 9m+15w
(9m+15w)/(3w) - (9m+15w)/(2m) = 5
18m^2 +30mw - 27mw - 45w^2 = 30mw
2m^2 -3mw - 5w^2 = 0
2m^2 + 2mw - 5mw - 5w^2 = 0
2m(m+w) - 5w(m+w) = 0
m:w = 5:2

Find the 28383rd term of the series: 123456789101112…

28383rd term
1 digit 9 terms
2 digits 90×2=180 terms
3 digits 900×3=2700 terms
4 digits 9000×4=36000 terms
So it is in the 4 digit number series
9+180+2700=2889
28383-2889=25494
25494/4=6373 with remainder 2
6373rd 4 digit number is 6373+999=7372... remainder 2 so 7373 and so, 3

There are three equal containers that are completely filled with different water-alcohol mixtures with water and alcohol in the ratio 2 : 3, 3 : 4 and 4 : 5 respectively. They are emptied into a bigger container. What fraction of the mixture in the bigger container should be replaced by water so that the resulting mixture has equal quantities of water and alcohol?

The keyword here is that all the containers are equal in volume. So, the first container will be in the form of 5x units, second 7x units and third 9x units. So, let the volume of each individual container be 3150 units. Total volume will be 9450 units.
Water content in the containers will be 1260 units, 1350 units and 1400 units respectively. Total volume of water will be 4010 units out of 9450 units.
If x units of the solution is removed and replaced by the same amount of water, we get the volume of water to be 4725 units. So, putting it in the form of an equation:
4010 - x*(4010/9450) + x = 4725
5440x=715*9450
x/9450 = 715/5440 as the fraction of the container is asked
x/9450 = 143/1088

Let A be a two-digit number and B be another two-digit number formed by reversing the digits of A.
If A + B + (Product of digits of the number A) = 145, then what is the sum of the digits of A?

Let the original number be 10a+b. The reverse will be 10b+a. So, 11a+11b+ab=145
11a+b(11+a)=145
11(11+a)+b(11+a)=266
(11+a)(11+b)=266
266=2 × 7 × 19
We need two factors of 266 each of whom are greater than 11. So 14 and 19. So the sum of digits will be 11.

The coordinates of the vertices of the triangle ΔABC are A(0, 1), B(4, 4) and C(5, 3). From which vertex the length of the median drawn to the opposite side is the minimum?
(1) A
(2) B
(3) C
(4) All medians are of equal length

The median from the vertex opposite to the longest side to the longest side will be of the least length among all three medians. As the longest side is AC, the answer is B.

When ‘A’ and ‘B’ are working alone, they can complete a piece of work in 25 days and 30 days respectively. On day 1, ‘B’ started the work and ‘A’ joined ‘B’ from day 3 onward. After how many days will the work be completed?

Let total work be 750 units. A does 30 units a day, B does 25 units a day. B works for 2 days and finishes 50 units. Then A and B work together and finish the remaining 700 units at the speed of 55 units per day. So, an additional 700/55=140/11=12.xx days. Total of 14.xx days.

A rhombus is drawn on the x-y plane by joining the points having coordinates A (0, 0), B (x1, y1), C (4, 6), D (x2, y2). What is the equation of the diagonal BD?
(1) 3x – 4y = 3
(2) 2y + 3x = 1
(3) 3y + 2x = 13
(4) 6x – 2y = 1

3 is correct. Diagonals of a rhombus are perpendicular bisectors of each other and so, midpoint of AC will lie on BD as well. So, we can simply substitute (2,3) in the equations and get the answer.

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