SAT Section 2 Math

Explanation
Since M N O P andQ R O S are squares, M N = N O = 8 and Q R = R O. Thus, M U = N R and the rectangle with area x^2 must be a square with sides of length x. The rectangles with area 5 times x must have sides of length x and 5. Thus, the area of Q R O S is equal to 5^2 = 25.

Another approach is the following. Since Q R O S is a square, Q R = R O. Thus, M U = N R and M U Q T is a square with sides of length x. The rectangles with area 5 times x must have sides of length x and 5. Since the area of M N O P is 64:



(x + 5)^2 = 64

x^2 + (5 times x) + (5 times x) + 25 = 64



Thus, the area of Q R O S is 25.

Explanation
The first step is to determine the area of the garden. Since the circumference is 20 feet, the radius is 20 over (2 times pi) = 10 over pi and the area is pi times (r^2) or pi times ((10 over pi)^2) = 100 over pi square feet. If there are 4 marigolds per square foot, 4 times (100 over pi) marigolds are needed. To find out how many packs of 6 flowers are needed, divide 400 over pi by 6. The result is 21.22; thus, 22 packs would be needed. The correct answer is E.

Explanation
It is often helpful to make a sketch for this type of question.

(Graphic)

The slope of line l is equal to negative 5 minus (negative 2) over (0 minus 6) = negative 3 over negative 6 = 1 over 2.
Since negative 5 is the y-intercept, the equation of l is y = ((1 over 2) times x) minus 5.

Since line p is perpendicular to line l, the slope of p is equal to negative 2 (the negative reciprocal of the slope of l ). Since line p contains the point with coordinates(negative 5 comma 0), the equation of line p can be found by substitution.

Stacked Equation

To find the x coordinate of the point where the two lines intersect, solve the equation ((1 over 2) times x) minus 5 = negative (2 times x) minus 10. The lines intersect at the point where x = negative 2.

It is also possible to graph the two linear equations to determine where they intersect. Using a graphing calculator, let y_1 = ((1 over 2) times x) minus 5 and y_2 = negative (2 times x) minus 10. You can find the point of intersection graphically in the standard viewing window, or you can examine a table of values to find the x-value where y_1 = y_2.

Explanation
One way to solve this problem is to set up an equation using one variable. Let n represent the number of 3 dollars tickets sold; then 50 minus n represents the number of 5 dollars tickets sold.


(3 times n) + (5 times (50 minus n)) = 230

(3 times n) +250 minus (5 times n) = 230

20 = 2 times n

10 = n

There were 10 3 dollar tickets sold.

Simultaneous equations can also be set up to solve the problem. Let n represent the number of 3 dollar tickets sold and p represent the number of 5 dollar tickets sold.


n + p = 50
(3 times n) + (5 times p) = 230

Another way to solve this problem is to realize that each of the tickets costs at least 3 dollars. At 3 dollars each, a total of 150 dollars would be collected from selling all 50 of the tickets. Since the remaining 80 dollars was the extra 2 dollars collected from each of the 5 dollar tickets, 40 of the 5 dollar tickets were sold. This implies that 10 of the 3 dollar tickets were sold.

Explanation
For this question, it may help to draw a diagram.

Graphic

The line y = 7 intersects the circle in 2 points. These two points and the center of the circle form a triangle with two sides of length 3 (since the radius of the circle is 3) and with height 2. (The distance between the point (4 comma 5) and the line y = 7 is the distance between (4 comma 5) and (4 comma 7).)

Graphic

Using the Pythagorean theorem, you can find that A B = square root 5. Point B has x-coordinate 4. Therefore, the x-coordinate of the points of intersection are 4 minus square root 5 and 4 + square root 5 or 1.76 and 6.24. The correct answer is D.

You could also solve the problem algebraically. A circle with center (4 comma 5) and radius 3 has equation (x minus 4)^2 + (y minus 5)^2 = 9. Substitute y = 7 into the equation, and it simplifies to (x minus 4 ) ^2 = 5. Solving for x produces the two x-coordinates of the points of intersection of the circle and the line.

Explanation
297/3 = 99 ; 99/3 = 33 ; 33/3 = 11. So, n = 3.
Note 297/(33) = 297/27 = 11, whereas 297/(34) = 297/81 = 3.67.

Explanation
Mr. Winkle is presently (q + r) years old.
Therefore, he will be (p + q + r) years old p years from now.
If the problem appears confusing, choose an easy set of numbers,
e.g., p = 7; q = 50 and r = 3.
Then, Mr. Winkle was 50 years old 3 years ago.
Therefore, he is 53 and will be 60 years old 7 years from now.
Note that p + q + r = 7 + 50 + 3 = 60.

Explanation
Let the distributed amounts be m, 2m, 3m and 4m. Then,
m + 2m + 3m + 4m = 35000 or 10m = 35000. Thus, m = 3500 or 4m = 14000.

Explanation
Let there be y caps in the box. Then, there are 2y/5 red caps.
One-fourth of the caps are blue. So, there are y/4 blue caps.
Therefore, number of green caps = y − 2y/5 − y/4 =(20y − 8y − 5y)/20 = 7y/20.
Ratio of green caps to blue caps = (7/20) ÷ (1/4) = (7/20) × 4 = 7/5

Explanation
Let m be your marks in the first quiz.
Then, your marks are 0.4m in the second quiz and m in the third quiz.
Percent Increase
= (Marks in third Quiz - Marks in second quiz) × 100/Marks in second quiz
= (m − 0.4m) × 100 / 0.4m = (0.6/0.4) × 100 = 150.

Explanation
Let the largest of the four consecutive even integers be m. Then,
m + (m − 2) + (m − 4) + (m − 6) = 892 or 4m − 12 = 892.
Thus, m = 904 / 4 = 226.
The four consecutive even integers are 220, 222, 224 and 226.

Explanation
Adding the equations gives 7y + 7z = 49.
Dividing by 7 gives y + z = 7.
So, Arithmetic Mean = (y + z)/2 = 7/2 = 3.5.

Explanation
Percent of false questions = (3/7) × 100 = 42.86.
So, about 43% of the questions are false.
Alternatively, one could reason out without any calculations as follows.
For every 4 true questions, there are 3 false questions. So, there are fewer false questions than true ones, i.e., less than 50% of the questions are
false. The only option is 43%.

Explanation
Now, 1001 is divisible by neither 3 nor 5.
So, let's try to divide by 7. We find 1001 / 7 = 143 = 13 × 11.
Since 1001 = 13 × 11 × 7, the smallest prime factor is 7.

Explanation
Now, (2.5/100) z = (75/100) 50. So, z = (75/2.5) 50 = (750/25) 50 = 1500.
Alternatively, 75 is 30 times 2.5. So, z must be 30 times 50.
Of the given options, only 1500 is large enough.
To save time, you can sometimes guess the answer without doing detailed computations.

Explanation
Percent Commission = (0.40/50) × 100 = 40/50 = 0.8.
Alternatively, 1% of $50 is 50 cents. Since the commission is 40 cents, it is less than 1%. So, the only option is 0.8%.

Explanation
Let the smallest of the four consecutive odd integers be m. Then,
m + (m + 2) + (m + 4) + (m + 6) = 968 or 4m + 12 = 968.
Thus, m = 956 / 4 = 239.
The four consecutive odd integers are 239, 241, 243 and 245.

Explanation
Let b be the side of the square and d the length of its diagonal.
Then, by Pythagoras theorem, d2 = b2 + b2 = 2 b2.
Since the diagonal of the larger square is 3 times that of the small square,
the side of the larger square is 3 times the side of the small square.
If b is the side of the small shaded square, then its area = b2.
Now, 3b is the side of the larger square and its area = 9b2.
Thus, unshaded area = 8b2 and
fraction of the area of the larger square unshaded = 8/9.

Explanation
The arithmetic mean of n numbers is their sum divided by n.
So, (3 + 5 + 7 + y) / 4 = 100 or 15 + y = 100 × 4 = 400.
Thus, y = 400 − 15 = 385.
Given that 3, 5 and 7 are small numbers and the average 100 is relatively large, it is possible to guess the answer as 385.

Explanation
2% of 7 = (2/100) × 7 = 14/100 = 0.14. So, 2% of 7% = 0.14%.