1. A certain type of mixture is prepared by mixing brand A at Rs.9 a kg. with brand B at Rs.4 a kg. If the mixture is worth Rs.7 a kg., how many kgs. of brand A are needed to make 40kgs. of the mixture?

There is a shortcut to this % of A=7-4=3% of B=9-7=2A=40(3/5)Brand A needed is 24kgs

2. A wizard named Nepo says "I am only three times my son's age. My father is 40 years more than twice my age. Together the three of us are a mere 1240 years old." How old is Nepo?Son's age = ANepo's age = 3AFather's age = 2(3A)+40 = 6A + 40The total isA + 3A + 6A + 40 = 124010A = 1200A = 1203(A)= 3(120) = 360Nepo is 360 years old.

3. A man ate 100 bananas in five days, each day eating 6 more than the previous day. How many bananas did he eat on the first day?let "x" = no of bananas eat at the first day.100=x+(x+6)+(x+12)+(x+18)+(x+24)100=5x+(60)100-60=5x40=5x8=x. .x=8

4. The minute hand of a clock overtakes the hour hand at intervals of 64 minutes of correct time. How much a day does the clock gain or lose?First calculate the interval of time it should take the minute hand to overtake the hour hand.

Letx = distance traveled by hour hand1 + x = distance traveled by minute hand

Time = Distance/Ratex/(1/12) = (1 + x)/1x = (1/12)(1 + x) = 1/12 + x/12(11/12)x = 1/12x = 1/11

Calculate time for minute hand to travel a distance 1 + x.

Time = Distance/Rate

(1 + x)/1 = 1 + x = 1 + 1/11 = 12/11 hour

Actual time to overtake on clock is 64 minutes.

64 min = 64/60 hours = 16/15 hour

Correct Clock Speed/Actual Clock Speed(12/11) / (16/15) = 12*15 / (11*16) = 180/176 = 45/44

The clock gains:45/44 - 1 = 1/44 day = 24/44 hour = 6/11 hour = 360/11 min= 32 min and 43 4/11 sec

5. Solve for x and y: 1/x - 1/y = 1/3, 1/x2 + 1/y2 = 5/9.x = 3/2 or -3 and y = 3 or -3/2.take 1/x=a, 1/y=b so, a-b=1/3, a2+b2=5/9, a2+b2=(a-b)2+2ab so, (a-b)2+2ab=5/9 (1/3)2+2ab=5/9 it gives ab=2/9 a=2/9b, so 2/9b-b=1/3 from this b=1/3 hence a=2/3 so x=3/2, y=3

6. A garrison of 3300 men has provisions for 32 days, when given at a rate of 850 grams per head. At the end of 7 days a reinforcement arrives and it was found that now the provisions will last 8 days less, when given at the rate of 825 grams per head. How, many more men can it feed?Sol. The problem becomes:3300 men taking 850 gms per head have provisions for (32 - 7) or 25 days, How many men taking 825 gms each have provisions for 17 days? Less ration per head, more men (Indirect Proportion)Less days, More men (Indirect Proportion)

Ration 825 : 850Days 17: 25 } : : 3300 : x(825 x 17 x x) = 850 x 25 x 3300 or x = (850 x 25 x 3300)/(825 x 17)=5000Strength of reinforcement = (5500 - 3300) = 1700.

(OR)

After 7 days = 3300*25*850 = Xso # men = (X/825)/(25-8) = 85000/17 = 5000so 1700 more men can be fed.

7. Three pipes, A, B, & C are attached to a tank. A & B can fill it in 20 & 30 minutesrespectively while C can empty it in 15 minutes. If A, B & C are kept open successively for 1 minute each, how soon will the tank be filled?

As with all pipes open it ll fill up 1/20+1/30-1/15 = 1/60 per minSo, to fill the tank it ll take 60 min.

8. What is the sum of all numbers between 100 and 1000 which are divisible by 14 ?14*8=11214*71=994therefore there are 64 (71-7) numbers between 100 and 1000 which are divisible by 14

since all these numbers are in arithmetic progressionsum=64/2*(112+994)= 35392

9. If s(a) denotes square root of a, find the value of s(12+s(12+s(12+ ...... upto infinity.4 Explanation : Let x = s(12+s(12+s(12+..... ) We can write x = s(12+x). i.e., x2 = 12 + x. Solving this quadratic equation, we get x = -3 or x=4. Sum cannot be -ve and hence sum = 4..

(OR)

let t = s(12 + s(12 + .... upto infinity=> t= s(12 + t)

or t^2 = 12 + t

or t^2 -t -12 = 0

or t^2 - 4t + 3t -12 = 0

or t(t - 4) +3 (t -4 ) = 0

or (t+3)(t-4) = 0

or t = -3 or t = 4

generally we are interested in positive square roots so t = 4but if we are taking -ive square roots also then t can be = -3