PROBLEM SOLVING AND DATA ANALYSIS OF SAT

Problem 1 :

Find three consecutive odd integers such that the sum of the first two is four times the third.

A) 3, 5, 7           B) -3, -1, 1     C) -11, -9, -7

D) -7, -5, -3       E) 9, 11, 13

Solution :

Let x be the odd integer, then x, x + 2 and x + 4 be three consecutive odd integers.

x + x + 2 = 4(x + 4)

2x + 2 = 4x + 16

2x - 4x = 16 - 2

-2x = 14

x = -14/2

x = -7

x, x + 2, x + 4 = -7, -7 + 2, -7 + 4

= -7, -5, -3

So, option (D) is correct.      

Problem 2 :

Find the shortest side of a triangle whose perimeter is 64, if the ratio of two of its sides is 4: 3 and the third side is 20 less than the sum of the other two.

A) 6     B) 18       C) 9       D) 10      E) 12

Solution :

Let first side = 4x

Second side = 3x.

Third side = (4x + 3x - 20)

4x + 3x + (4x + 3x - 20) = 64

14x - 20 = 64

14x = 64 + 20

14x = 84

x = 84/14

x = 6

First side = 4x = 4(6) = 24

Second side = 3x = 3(6) = 18

Third side = (4x + 3x - 20) = 4(6) + 3(6) - 20

= 24 + 18 - 20

= 22

The shortest side of a triangle is 18.

So, option (B) is correct.      

Problem 3:

A purse contains 16 coins in dimes and quarters. If the value of the coins is $2.50, how many dimes are there?

A) 6      B) 8      C) 9      D) 10      E) 12

Solution :

Let x be the number of dimes.

1 dimes = $0.10

1 quarter = $0.25

Number of quarters = 16 - x 

0.10x + 0.25(16-x) = 2.50

0.10x + 4 - 0.25x = 2.50

-0.15x + 4 = 2.50

-0.15x = 2.50 - 4

0.15x = 1.5

x = 10

Number of dimes is 10.

So, option (D) is correct.      

Problem 4 :

Danny drove to Yosemite Park from his home at 60 miles per hour. On his trip home, his rate was 10 miles per hour less and the trip took one hour longer. How far is his home from the park?

A) 65 mi      B) 100 mi     C) 200 mi

D) 280 mi    E) 300 mi

Solution :

Let x be the distance to be covered.

His original speed = 60 miles per hour

Time = Distance / Speed

Time taken = x/60

When he is driving 10 miles per hour less than the original speed, he will reach the destination one hour later.

x/50 - x/60 = 1

6x - 5x = 300

x = 300

So, the distance is 300 miles.

Problem 5 :

Two cars leave a restaurant at the same time and travel along a straight highway in opposite directions. At the end of three hours they are 300 miles apart. Find the rate of the slower car, if one car travels at a rate 20 miles per hour faster than the other.

A) 30     B) 40     C) 50     D) 55     E) 60

Solution :

Let x be the distance covered by the first car. Then 300 - x will be the distance covered by the second car.

Speed of both cars add upto = 300/3 ==> 100 miles.

Speed of first car + (Speed of first car + 20) = 100

2(Speed of first car) = 80

Speed of the slower car = 40 miles.

Problem 6 :

The numerator of a fraction is one half the denominator. If the numerator is increased by 2 and the denominator is decreased by 2, the value of the fraction is 2/3. Find the numerator of the original fraction.

A) 4    B) 8     C) 10     D) 12     E) 20

Solution :

Let the fraction be x/y.

x = (1/2)y

y = 2x ---> (1)

If the numerator is increased by 2 and the denominator is decreased by 2 and the denominator is decreased by 2, the value of the fraction is 2/3.

(x + 2) / (y - 2) = 2/3 ---> (2)

Put value of y from (1) in (2), we get

(x + 2) / (2x - 2) = 2/3

3(x + 2) = 2(2x - 2)

3x + 6 = 4x - 4

3x - 4x = -4 - 6

-x = -10

x = 10

The numerator of the original fraction is 10.

So, option (C) is correct.      

Problem 7 :

Darren can mow the lawn in 20 minutes, while Valerie needs 30 minutes to do the same job. How many minutes will it take them to mow the lawn if they work together?

A) 10     B) 8      C) 16      D) 6 1/2     E) 12

Solution :

Let x be the total work done.

Work done by Darren = x/20

Work done by Valerie = x/30 minutes

x/20 + x/30 = 1

5x/60 = 1

x = 12

x = 12 minutes

So, option (E) is correct.      

Problem 8 :

Meredith is 3 times as old as Adam. Six years from now, she will be twice as old as Adam will be then. How old is Adam now?

A) 6     B) 12    C) 18     D) 20    E) 24

Solution :

Let x be an Adam’s age now.

3x = Meredith’s age now

x + 6 = Adam’s age in 6 years

3x + 6 = Meredith’s age in 6 years

3x + 6 = 2(x + 6)

3x + 6 = 2x + 12

3x - 2x = 12 - 6

x = 6

So, option (A) is correct.      

Problem 9 :

Mr. Barry invested some money at 5% and an amount half as great at 4%. His total annual income from both investments was $210. Find the amount invested at 4%.

A) $1000    B) $1500     C) $2000      D) $2500    E) $3000

Solution :

Let x = amount invested at 4%

2x = amount invested at 5%

0.04x + 0.05(2x) = 210

Multiply 100 on both sides.

4x + 5(2x) = 21000

14x = 21000

x = 21000/14

x = $1500

The amount invested at 4% is $1500.

So, option (B) is correct.