NMC Sample Problems: Grade 8

Transcription

1 NM Sample Problems: Grade 8. right triangle has lengths of and 8 in sides. What is the length of hypotenuse (the longest side)? (b) 0 5. n isosceles triangle has the base length of 6 and each side of length. What is the area of the triangle? 9 5 (b) cone has the base circle with the radius of and the height of 6. What is the volume of the cone? 6π (b) π 6π π π. Suppose you are randomly taking two marbles from a bag of marbles with red, yellow and black. What is the probability that you picked one red and one black marbles? (b) 5. The boiling temperature of water is at 00 degrees in elsius. What is the boiling temperature in Fahrenheit? 00 (b) line intersects two points (, ) and (, ). Which one of the following expressions represents the line? y = x (b) y = x + y = x y = x y = x + nswer:. Solve for t in + 8t = 6t. t = (b) t = 5 t = t = 5 t = nswer: of 0

2 8. Solve for r in r + =. r = 5 (b) r = r = 0 r = 0 r = 5 nswer: 9. alculate (b) my plans to plant flowers and buys.5 pounds of potting soil. What is the minimum number of pots that my can fill when each pot can be filled with.5 pounds? (b) ngie, Bob, arlos, and Dan are randomly seated around a square table, one person to a side. What is the probability that ngie and arlos are seated opposite each other? (b). Two congruent squares, BD and P QRS, have side length 5. They overlap to form the 5 by 5 rectangle QRD as shown. What percentage of the area of rectangle QRD is shaded? (Figure not drawn to scale!) P B Q D S R 5 (b) There are 80 students at oldspring Middle School, where the ratio of boys to girls is 5:. There are 0 students at Warmspring Middle School, where the ratio of boys to girls is :5. The two schools hold a dance party and all students from both schools attend. What fraction of the students at the dance party are girls? 8 (b) nswer:. Let w, x, y, and z be whole numbers. If w x 5 y z = 588, what does w + x + 5y + z equal? (b) of 0

3 5. How many rectangles are in the figure below? 8 (b) If a b = a b a+b for a, b positive integers, then what is 5 0? 0 0 (b) 50 nswer:. The graph shows the price of five gallons of gasoline during the first ten months of the year. By what percent is the highest price more than the lowest price? Gas Price ($) Jan Feb Mar pr May Jun Jul ug Sep Oct 50% (b) 6% 0% 89% 00% 8. circle with radius is inscribed in a square and circumscribed about another square as shown. Which of the following is closest to the ratio of the shaded area in the circle to the total shaded area? 5 (b) of 0

4 9. Ken takes a long bike ride on a hilly highway. The graph indicates the miles traveled during the time of his ride. What is Ken s average speed for the entire ride in miles per hour? Miles Hours (b) nswer: 0. What is the sum of the mean, median, and mode of the numbers,,0,,,,0,? 6.5 (b) s my rides her bicycle on a long straight road, she spots Bob skating in the same direction a half mile in front of her. fter she passes him, she can see him in her rear view mirror until he is a half mile behind her. my rides at a constant rate of miles per hour, and Bob skates at a constant rate of 8 miles per hour. How many minutes have passed since my first spotted Bob until she lost him in the rear view mirror? 6 (b) nswer:. Father takes a bus to the ity for work, and buses are dispatched at every 0 minutes. ssuming that there are no bus delays, how long does your Father have to wait at a bus stop on the average if he comes to the bus stop without prior information on when buses left the bus depot? 5 (b) my s video game lets her design the appearance of her character. There are different bodies and faces to choose from. How many different characters can my design? (b) 0 nswer: of 0

5 . Six pepperoni circles will exactly fit across the diameter of a -inch pizza when placed as shown. If a total of pepperoni circles are placed on this pizza, what is the fraction of the pizza covered by pepperoni? 5 (b) The top of one tree is 6 feet higher than the top of another tree. The heights of the two trees are in the ratio :. In feet, how tall is the taller tree? 8 (b) Of the 500 balls in a large bag, 80% are red and the rest are blue. How many of the red balls must be removed from the bag so that 5% of the remaining balls are red? 5 (b) nswer:. The lengths of the sides of a triangle are three consecutive integers. The length of the shortest side is 0% of the perimeter. What is the length of the longest side? (b) nswer: 8. square and a circle have the same area. What is the ratio of the side length of the square to the radius of the circle? π (b) π π π π 9. decorative window has a rectangle with semicircles on either end. The ratio of D to B is :, and B = 0 inches. What is the ratio of the area of the rectangle to the sum of areas of two semicircles? D B : (b) : 6 : π 9 : π 0 : π 5 of 0

6 0. Two circles in the picture have the same center. The line D is tangent to the inner circle at B, is 0, and the line D has length 6. What is the area between the two circles? B D 6π (b) 9π 6π 8π 00π. In a room, /5 of all the people are wearing gloves, and / of the people are wearing hats. What is the minimum number of people in the room wearing both gloves and hats? (b) my bought a novel, and read /5 of the pages plus more on the first day. On the second day, she read / of the remaining pages, plus 5 pages. On the third day, she read / of the remaining pages, plus 8 pages. She then realized that there were only 6 pages left to read. How many pages are in this book? 0 (b) The hundreds digit of a three-digit number is more than the units digit. The digits of the threedigit number are reversed, and the result is subtracted from the original three-digit number. What is the units digit of the result? 0 (b) 6 8 nswer:. Semicircles P OQ and ROS pass through the center O of a circle. What is the ratio of the sum of the areas of the two semicircles to the area of the larger circle? P (, ) Q(, ) R(, ) S(, ) (b) π 6 of 0

7 5. Given three numbers 0 8, 5, and, which one is a correct ordering of the numbers? < 0 8 < 5 (b) < 5 < < < < 5 < 0 8 < < 5 6. In the design as shown, the smallest black circle has radius inches, with each successive circle s radius increasing by inches. pproximately what percent of the design is black? (b) In square BE, F = F E and D = DE. What is the ratio of the area of triangle BF D to the area of square BE? B F E D 6 5 (b) Ten tiles have numbers through 0 painted on them, and are placed face down. You randomly pick a tile and throw a die, and multiply the two numbers from a tile and the die. What is the probability that the product is a square? 0 (b) How many positive integer value of n can have both n/ and n three-digit whole numbers? (b) 0. onsider all rectangles with integer side lengths and a perimeter of 50 units. What is the difference between the largest and smallest areas of the rectangles? 6 (b) of 0 nswer:

8 . Three s, three Bs, and three s are placed in the nine spaces so that each row and column contain one of each letter. If is placed in the upper left corner, how many such arrangements are possible? (b) 5 6. Given three boxes, you weigh pairs of them in every possible way with weights of, 5, and pounds. What is the combined weight of the three boxes? 60 (b) ball is dropped from a height of meters. On its first bounce, it rises to a height of meters. It keeps falling and bouncing back up to / of the height it reached in the previous bounce. On which bounce will it first not rise to a height of 0.5 meter? (b) 5 6. The average age of 6 people in room is 0. The average age of people in room B is 5. If the two groups are combined, what is the average age of the combined 0 people?.5 (b).5 5 nswer: 5. The investment of $00 suffered a 5% loss in the first year. In the nd year, however, the remaining investment after the first year showed a 0% gain. Over the two-year period, what is the change in the investment? 5% loss (b) % loss % gain % gain 5% gain nswer: 6. In how many ways can 0,00 be written as the sum of two primes? 0 (b) 8 of 0

9 . On the dart board, the outer circle has radius 6 and the inner circle has radius. Three radii divide each circle into three congruent regions, with point values shown in the figure. The probability that a dart hits a given region is proportional to the area of the region. When two darts hit this board, the score is the sum of the point values in the regions. What is the probability that the score is odd? (b) 6 8. Suppose you have five circles with radii,,,, and 5. Which of the graphs will plot the point (, ), where is the circumference and is the area of the five circles. (b) 9. For any positive integer n, define n to be the sum of all positive factors of n. For example, 6 = =. Find. (b) nswer: 50. grid must have each of digits through once in each row and once in each column. What number will occupy the lower right-hand square in the grid? (b) annot be determined 9 of 0

10 5. There are four consecutive integers such that the sum of the cubes of the first three numbers equals the cube of the fourth number. Find the sum of the four numbers. nswer: 8 5. point (x, y) is randomly chosen such that 0 x and 0 y. What is the probability that y x? nswer: 5. Find the smallest three-digit number whose remainders are,, and 5 when it is divided by,, 5 and 6, respectively. nswer: 9 0 of 0