Problem Solving? and necessary solutions..?

Question

1. Find the sum of all multiples of 9 between 1 and 400.

2. The denominator of a fraction is two less than the numerator. If the numerator is decreased by 1 and the denominator is doubled, the sum of the original fraction and the new fraction is 7/3. Find the original fraction.

3. Admission tickets to a concert cost P200 for students and P300 for adults. If the total revenue amounted to P82,400 and twice as many adult tickets as student tickets were sold, how many tickets of each type were sold?

4. If two pipes can fill a pool in 3 hours when both are open and one of the pipes can fill the pool alone in 5 hours, how long would it take the other pipe to fill the pool alone?

5. Determine the speed of a plane in still air and the speed of the wind, if it takes 3 hours for the plane to travel a distance of 900 miles with the wind and 5 hours to travel the same distance if it is going against the wind.

Answer 1

1)

There are 44 multiples of 9 up to 400

1*9+2*9+3*9+........+44*9

=9(1+2+3....+44)

You can use the Gauss method  to calculate the sum that is

22 pairs of 44+1, 43+2, 42+3......

so it totals 22*45=990

9*990=8910

2)

x/(x-2)

and

(x-1)/[2(x-2)]

find the common demoninator of 2(x-2)

2x/[2(x-2)] + (x-1)/[2(x-2)] = 7/3

(2x+x-1)/[2(x-2)]=7/3

3x-1 = 14(x-2)/3

9x-3 = 14x-28

-5x = 25

so x=5

The original fraction was 5/3

3)

let number of students be 'x'

and number of adults be 'y'

200x+300y = 82 400

We also know

y=2x

as twice as many adult tickets were sold

substitute it into the equation and we get

200x+300(2x) = 82400

800x = 82400

x = 103

so 103 student tickets were sold and 206 adult tickets were sold.

4)

The first pipe fills the pipe at 1/5 of the pool per hour

the second fills it at 'x' of the pool per hour

if it takes 3 hrs to fill with both on, together they fill at 1/3 of the pool per hour

1/3 = 1/5 + x

x = 1/3 - 1/5

x = 2/15

it fills 2/15 of the pool every hour

so it will take 15/2 hours or 7.5hrs to fill the pool alone

5)

Let 'x' be the speed of the plane in still air and 'y' be the speed of the wind

(x+y)*3=900

(x-y)*5=900

solve using simultaneous equations

(x+y)*3 = (x-y)*5

3x+3y = 5x-5y

8y=2x

x=4y

substitute this into the first equation

(x+y)*3=900

(4y+y)*3=900

15y=900

y=60

x=240

The plane can travel at 240miles per hour in still air and the wind is at 60miles per hour

Answer 2

1. The highest mult. of 9 below 400 is 396=9*44. That means we have a total of 44 mults. of 9 in the interval.

The multiples of 9 are an arithmetic series. The sum of a series is:

S=(min+max)*n/2

S=(9+396)*44/2=405*22=8910

2.

d=n-2

(n-1)/2d + n/d = 7/3 =

= (n-1)/2d+2n/2d = (3n-1)/2d =

=(3n-1)/(2n-4)=7/3

9n-3=14n-28

5n=25

n=5

d=5-2=3

n/d=5/3

3.

20s+300a=82400

a=2s

figure it out.

4. 1-st pipe: 1/5 pool/hour

In three hours: 3/5 pool

2-nd pipe in three hours: 1-3/5 pool = 2/5 pool

2-nd pipe in one hour: (2/5)/3 pool = 2/15 pool

Amount of time to fill pool: 1/(2/15)=15/2=7.5 hrs.

5. p+w=900/3=300

p-w=900/5=180

Sum of the equations:

2p=480

speed of the plane: p=240

speed of the wind: w=300-p=60