math questions 7

1. Using your knowledge of geometry, explain what makes a rectangle and a parallelogram similar. Discuss the relationship between finding the area of each.

A parallelogram is a quadrilateral with two pairs of parallel sides. The opposite sides and opposite angles are congruent. The rectangle is a parallelogram with four right angles. Its adjacent sides are perpendicular. The rectangle has two axes of symmetry that pass through midpoints of the opposite sides. The area of a rectangle is A=LW and the area of a parallelogram is A=BH. They are similar because they both have two pairs of parallel sides, except the parallelogram has slanted sides. To find the area of a parallelogram you can also use the rectangle formula: A=LW by drawing a perpendicular line from the base to the opposite side. When drawing this perpendicular line you will see a right triangle. Cut this off the parallelogram and it then becomes the shape of a rectangle. Then, multiply the base (length) times the height (width). This will give you the answer.

8. Angela has nickels and dimes in her pocket. She has twice as many dimes as she has nickels. She has $4.35 in her pocket. How many nickels does she have? Be sure to show the algebraic formula that illustrates your logic.

Angela has 36 dimes and 17 nickels ($3.60 + $.85). Since the dimes are worth twice as much and there are twice as many, D (number of dimes) = 4/5 x ($4.35)/10. N (number of nickels) = (1/5 x 4.35)/5. Since there are at least twice as many dimes, round the dimes up to 36 and the nickels down to 17.

7. Which of the following shows 2,520 as the product of its prime factors? A. 2 x 3 x 5 x 72 B. 22 x 3 x 52 x 7 C. 22 x 33 x 5 x 7 D. 23 x 32 x 5 x 7

First make a factor tree for 2520 252 * 10 126 * 2 2 * 5 9 * 14 3 * 3 7 * 2 so 2520 = 2 * 2 * 2* 3* 3* 5 * 7 or 23* 32 * 5 * 7 The answer is D.

3. Separate 60 into two parts such that one part will exceed the other part by 36. (Use x for the larger, y for the smaller.) Show your work.

Let y = x - 36 x = larger # x + x-36 = 60 combine like terms 2x = 96 divide x = 48 y = 12

9. How many different two-digit numbers can be written if each digit must be one and only one of the integers from 1 to 8 inclusive?

There can be 8 first integer combinations (1,2,3,4,5,6,7,8) and each of those can have 7 second integer combinations 11, 12, 13, 14 etc so 8 x 7 = 56 combinations.

4. An artist is planning to construct a rectangular wall design from square tiles. The wall design is to be 72" long and 42" wide. All the square tiles must be the same size, and the length of the sides of the tiles must be a whole number.

Using your knowledge of number theory and geometry: • Find three different sizes of square tiles that could be used to completely fill the rectangular space, with no tiles overhanging the border; and • Determine the smallest number of square tiles that could be used to fill the rectangular area. The sides are 72 x 42 and tiles must be square, so you must find 3 numbers that are divisors of both 72 and 42. First, I know 2 will work because both are even #'s then I know 3 will work because the sum of both numerals in the number equal a multiple of 3 ( 7 + 2 = 9 and 4 + 2 = 6 ; 9 and 6 are both multiples of 3 ). Also I know 6 will work. So tiles can be 2" x 2", 3"x 3" and 6" x 6" Using 6 x 6 tiles, you would have sides of 12 tiles by 7 tiles. This would require 84 tiles to complete. Since 6" x 6" is the largest tile, 84 tiles would be the least amount of tiles needed.

2. Evaluate the following if a = 2 and b = -3: (2ab - 4b²) ÷ ab.

[(2 x -6) - (4 x 9)] ¸ (2 x -3) (-12 - 36) ¸ -6 -48 ¸ -6 = 8

10. There are 4 roads connecting Town X to Town Y and 3 other roads connecting Town Y to Town Z. Monica must drive from Town X to Town Z through Town Y every day to her job. What is the greatest number of consecutive days that Monica could make the trip without taking exactly the same route?

4 ways to get from X to Y and 3 ways to get from Y to Z. Therefore, there are 12 different ways to get from X to Z.