Math Tutoring Service

A poultry farm has only chickens and pigs. When the manager of the poultry counted the heads of the stock in the farm, the number totaled up to 200. However, when the number of legs was counted, the number totaled up to 540. How many chickens were there in the farm?

Let there by 'x' chickens and 'y' pigs.
Therefore, x + y = 200 --- (equation #1)
Each chicken has 2 legs and each pig has 4 legs
Therefore, 2x + 4y = 540 --- (equation #2)

Solving equations #1 and #2, we get x = 130 and y = 70. 

Three years back the age of a father was 24 years more than his son. At present the father is 5 times as old as the son. How old will the son be three years from now?

Let x be the age of the son three years ago,then the age of his father three years ago is 
x + 24,thus their present age is x + 3 for the son and x + 27 for the father, using the second 
premise of the problem, the father is 5 time as old as his son, we have an equation based 
on the present age, x + 27 = 5(x + 3), solving for x we have x = 3, which is the age of the 
son three years ago and the present age of the son is 6 years old thus the the son's age three 
years from now is 9 years old. 

For what values of ‘k’ will the pair of equations 3x + 4y = 12 and kx + 12y = 30 not have a unique solution?

For the two systems with no solution the systems should be inconsistent system, 
meaning the there is no intersection between the two line or they are parallel to 
each other, if two lines are parallel their slopes must be the same. the slope of 
3x + 4y = 12 is m1 = - 3/4 and the slope of kx + 12 y = 30 is m2 = - k/12, 
since m1 = m2, then - 3/4 = - k/12, solving for k = 9 

A train traveling at 72 kmph crosses a platform in 30 seconds and a man standing on the platform in 18 seconds. What is the length of the platform in meters?

Using D = rt, the rate of the train in meters per second 
(Transform the rate to m/s = 7200 m / 3600 seconds = 20 m/s) is 20 m/s to find 
for the total number of seconds the train pass the flatform we have to subtract 
the total number of time the train crosses the flatform and the time the man 
is standing on the flatform, that is 30 - 18 = 12, thus the length of 
the flatform is 20(12) = 240 meters 

A train traveling at 100 kmph overtakes a motorbike traveling at 64 kmph in 40 seconds. What is the length of the train in meters?

Change the rate of the train and motorbike to mps,
we have 10000 m / 3600 s = 27.78 mps for the train and 64000m / 3600 s = 17.78 mps 
for the motorbike, using the principle of overtaking, distance travelled by the train 
must be equal to the distance travelled by the motorbike, that is DTrain = Dmotorbike, 
the distance travelled by the train is D = (27.78)(40) = 1111.2 meters and the distance 
travelled by the motorbike is D = (17.78)(40) = 711.2, and since the distance 
travelled by the train and the motorbike must be the same, to get for the length 
of the train, we have 1111.2 - 711.2 = 400 meters, thus the length 
of the train is 400 meters.  

Jim travels the first 3 hours of his journey at 60 mph speed and the remaining 5 hours at 24 mph speed. What is the average speed of Jim’s travel in mph?

Solving=for the average speed of Jim using partial distance travelled,
3(60) = 180 and 5(24) = 120, the total distance travelled is 180 + 120 = 300 m divided by the total
number of hours which is 3 + 5 = 8, thus the average speed of Jim is 300 m divided by
8 hours = 37.5 mph  

Three friends Alice, Bond and Charlie divide 1,105 dollars amongst them in such a way that if 10, 20 and 15 are removed from the sums that Alice, Bond and Charlie received respectively, then the share of the sums that they got will be in the ratio of 11 : 18 : 24. How much did Charlie receive?

1,105 - (10 + 20 + 15) = 1,060, and since the ratio after the 
deduction is 11:18:24, we have an equation 11x + 18x + 24x = 1060 and x = 20, thus Charlie 
received 24(20) = 480 + 15 = 495  

Mary and Mike enter into a partnership by investing P700 and P300 respectively. At the end of one year, they divided their profits such that a third of the profit is divided equally for the efforts they have put into the business and the remaining amount of profit is divided in the ratio of the investments they made in the business. If Mary received P800 more than Mike did, what was the profit made by their business in that year?

Since Mary and Mike invest 700 and 300 respectively the total number 
of investment is P1000, let x be the rate of the investment and using the formula I = PRT, 
the total profit is equal to I = (1000)(x)(1) = 1000x and 1/3 of that is 1000x/3 and divided 
to them equally we have 1000x / 6 and the remaining 2/3 of 1000x is divided by the ratio 
between the investments of each that is 1000x (7/15) for Mary and 1000x (1/5) for Mike, 
since Mary received 800 more than Mike we have an equation 
(1000x / 6) + 1000x (7/15) = (1000x / 6) + 1000x (1/5) + P800, solve for x = 3, 
thus the total profit is 3000. 

In what ratio should a 20% methyl alcohol solution be mixed with a 50% methyl alcohol solution so that the resultant solution has 40% methyl alcohol in it?

The ratio is 1:2, using the guess and check strategy we 
have 20%(1) + 50%(2) = 40%(1 + 2), 1.2 = 1.2 

What is the highest integral value of 'k' for which the quadratic equation x2 - 6x + k = 0 have two real and distinct roots?

To have two real roots and distinct roots the discriminant of 
the quadratic function must be greater than zero, that is b2 - 4ac > 0, 
thus (- 6)2 - 4(1)(k) > 0, the k < 9, thus the highest integral 
value for k is 8.  

If one of the roots of the quadratic equation x2 + mx + 24 = 0 is 1.5, then what is the value of m?

Since 1.5 is a root of the quadratic equation, substituting it 
to the equation (1.5)2 + 1.5 m + 24 = 0, solving for m we have - 17.5   

For what value of ‘m’ will the quadratic equation x2 – mx + 4 = 0 have real and equal roots?

To have a real root and equal the discriminant must be equal to zero, 
that is b2 - 4ac = 0, then m2 - 4(1)(4) = 0, m2 = 16, taking the square 
root we have m = - 4, and + 4