KSEA NMC Sample Problems Grade 7 and 8. (A) 9 (B) 3 (C) 0 (D) 3 (E) None of the above. If a quadrilateral is not a rhombus, then it is not a square.

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1 1. What is the product of all valid solutions of the fractional equation x 45 5x = 1 5(x 1)? (A) 9 (B) 3 (C) 0 (D) 3 (E) None of the above 2. Use the statement below to answer the question that follows. If a quadrilateral is not a rhombus, then it is not a square. If the statement above is true, then which of the following statements must be true? (A) If a quadrilateral is not a square, then it is a rhombus. (B) If a quadrilateral is a square, then it is a rhombus. (C) If a quadrilateral is a rhombus, then it is a square. (D) If a quadrilateral is not a square, then it is not a rhombus. (E) If a quadrilateral is a rhombus, then it is not a square. 3. Find the value of (A) (B) 1 (C) (D) 4 (E) If three numbers are randomly selected without replacement from 0, 1, 2, 3, and 4, how many three digit numbers can we make? (A) 125 (B) 64 (C) 60 (D) 48 (E) The graph of a function f(x) = a passes through two points (8,2) and (b,2b). Find x2 the sum of a and b. (A) 4 (B) 32 (C) 68 (D) 128 (E) 132 1

2 6. If Brittany gets a 97 on her next math test, her average will be 89. However, if she gets 72, her average will be 84. How many tests has Brittany already taken? (A) 3 (B) 4 (C) 5 (D) 7 (E) 8 7. We partition the rectangle ABCD into seven congruent rectangles like the figure below. If the area of the ABCD is 84 cm 2, what is the perimeter of a small rectangle in cm? (A) 9 (B) 7 (C) 14 (D) 19 (E) Tom plants 5 white rosebushes and 2 red rosebushes in a row. How many different ways can he plant at least four white bushes consecutively? (A) 21 (B) 4 (C) 6 (D) 9 (E) Two circles, A and B, overlap each other as shown below. The area of the common part is 2 5 of the area of Circle A, and 5 of the area of Circle B. What is the ratio of 8 the radius of Circle A to that of Circle B? (A) 2 : 1 (B) 3 : 2 (C) 4 : 3 (D) 5 : 4 (E) 6 : 5 2

3 10. Six students take a photo for a yearbook sitting in a row. Two of them wear hats, and they want to sit at both ends. How many different ways of seating all the students are possible? (A) 24 (B) 48 (C) 60 (D) 120 (E) Seven individual socks are drawn at random without replacement from a basket containing seven different pairs of socks. If the first six socks drawn are all different from one another, what is the probability that the next sock drawn at random from the basket is also different from the rest? (A) 1 2 (B) 1 4 (C) 1 7 (D) 1 8 (E) Find all value(s) of y which satisfy the following equations. y = 1 2(x+2) and x 2 +3x 4 = 0 (A) 1 (B) 3 (C) 5 (D) 1 and 3 (E) 3 and Which of the following number is the smallest when p < 1 and 0 < q < 1? (A) 1 p (B) 1 q (C) p (D) p (E) q 14. Amy and Bryan walk toward each other fromato B. Amy walks at 220 feet per minute and Bryan takes 15 minutes to walk a mile. The distance between A and B is 11 miles. If they start walking to each other at the same time, approximately how long will it take for them to meet? (Hint: 1 mile = 5280 feet) (A) 40 minutes (B) 1 hour (C) 1 hour 20 minutes (D) 1 hour 42 minutes (E) 2 hours 3

15. Complete the proof. Consider a circle(center is O) that inscribes the triangle RST and has the diameter ST. We want to show that RST = 1 2 ROT. Proof: We begin by constructing radius OR.

4 15. Complete the proof. Consider a circle(center is O) that inscribes the triangle RST and has the diameter ST. We want to show that RST = 1 2 ROT. Proof: We begin by constructing radius OR. Since ROT is an exterior angle of ROS, ROT = (1). Since the radius of the circle is OR = OS, ROS is (2). Since ROS is (2), ORS = (3). Thus, RST = 1 2 ROT. (1) (2) (3) (A) ORS isosceles OSR (B) ORS + OSR isosceles OSR (C) ORS + OSR equiangular OSR (D) ORS isosceles OSR (E) ORS + OSR isosceles ROS 16. Evaluate (A) 1,500,500 (B) 499,999 (C) 500,999 (D) 500,500 (E) 499,500 4

5 17. If it was two hours later, it would be half as long until midnight as it would be if it was an hour later. What time is it now? (A) 7:00 PM (B) 8:00 PM (C) 9:00 PM (D) 8:30 PM (E) 9:30 PM 18. The interior angles of a quadrilateral inscribed in a circle are α, β, γ and δ as shown in the figure. Which of the following is always true? (A) α+β +γ +δ = 180 (B) γ +δ = 90 (C) α+β = 90 (D) β +δ = 90 (E) α+γ = β +δ 19. The Fine Art Center charged an admission fee of $5 per adult and $3 per child. On April 15, the center earned $1350 in total. On the next day, the Center earned $1365 and had the number of adults decreased by 5% and the number of children increased by 6%. How many children did the Center have on April 15? (A) 90 (B) 114 (C) 120 (D) 250 (E) If x and y are distinct prime numbers, which of the following numbers must be odd? (A) xy (B) 2(x+1) 2y +1 (C) x+y +3 (D) 3xy 1 (E) 2x+y 5

6 21. A spherical balloon is inflated to a given diameter. If the diameter is then doubled, what is the ratio of the new volume to the original volume? (A) 1 8 (B) 1 4 (C) 1 2 (D) 2 1 (E) When the solution set of the inequality 5 1 2x 3 12 is {x a x b}, find the product of a and b. (A) 4 (B) 2 (C) 0 (D) 2 (E) If a b = c where abcd 0, which of the following statements is NOT always true? d (Assume that all denominators are non equal to zero.) (A) a c = b d (D) a d a+d = c d c+d (B) a+2b b (E) 3a b 5b = c+2d d = 3c d 5d (C) a a+b = c c+d % of the families in Tombs county subscribe to Time, 50% of the families subscribe to Newsweek, and 30% of the families subscribe to Economist. 10% of the families subscribe to both Time and Newsweek, 10% of the families subscribe to both Time and Economist, 15% of the families subscribe to both Newsweek and Economist, and 5% of the families subscribe to all three. What percentage of the families in Tombs county subscribe to only one of the three magazines? (A) 45% (B) 55% (C) 65% (D) 75% (E) 80% 25. Katie, Sunny, and 3 other boys make a team for a quiz show. In how many different ways can 5 people be seated in a row if Katie and Sunny sit next to each other? (A) 24 (B) 30 (C) 48 (D) 52 (E) 60 6

7 26. A function f has the following properties. What is the value of f(243)? f(3) = 1 and f(xy) = f(x)+f(y) for any real numbers x and y (A) 15 (B) 12 (C) 10 (D) 5 (E) If the width of a particular rectangle is doubled and the length is increased by 4, then the area is tripled. What is the original length of the rectangle? (A) 4 (B) 6 (C) 8 (D) 10 (E) The lengths of two sides of a triangle are 2 and 9. Which of the following could be the length of the third side? (A) 3 (B) 6 (C) 7 (D) 8 (E) Simplify (A) 9 90 (B) 9 33 (C) 3 23 (D) 3 61 (E) If a square has two of its vertices at (2,2) and (2, 2), how many such squares are possible? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 7

8 31. Which of the following is equivalent to the expression? ( ) x p 2 ( x 3p ) q y 2q y p (A) x 2q 2 y 4 p (D) x p(q 2) y qp 4q (B) x 3q 2p y 2q (E) x p(3q 2) y q(4 p) (C) x 3q 2p y 4q Which of the following is correct about l : y = a 1 x+c 1 and m : y = a 2 x+c 2 where a 1, a 2,c 1, and c 2 are constants? (A) a 1 a 2 = 0 (B) a 1 +c 1 < 0 (C) a 2 +c 2 > 0 (D) a 1 c 2 +a 2 c 1 < 0 (E) a 1 a 2 +c 1 c 2 > You have been given 10 weights in the same shape and color, one of which is slightly heavier than the rest. Using a balance, what is the smallest number of weighings that you can always use to find out the heavier weight? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 8

34. Let N(x) bethenumber ofpositive integersy such thaty x; forexample, N(5) = 2 since only 1 and 2 are positive integers less than 5. Find the value of N(1)+N(2)+ +N(10).

9 34. Let N(x) bethenumber ofpositive integersy such thaty x; forexample, N(5) = 2 since only 1 and 2 are positive integers less than 5. Find the value of N(1)+N(2)+ +N(10). (A) 55 (B) 30 (C) 25 (D) 21 (E) Each square in the partially filled grid shown contains a number from 1 to 25, each number appearing exactly once. The numbers in each row and each column add up to 65. What is the value of x+y? (A) 15 (B) 17 (C) 19 (D) 21 (E) In sequence 1,2,5,10,17,26,37,50,, find the 50th number of the sequence. (A) 2400 (B) 2401 (C) 2402 (D) 2501 (E) Suppose that x and y are positive integers and y 2 x 2 = 143. Find one possible value of x+y. (A) 11 (B) 13 (C) 15 (D) 17 (E) 19 9

10 38. Consider a sequence of numbers, A 0,A 1,A 2,. The entries of the sequence satisfy the following properties A 0 = 1, and A 1 = 2, Find the sum A 1 +A 2 +A 3 + +A 100. A n+1 = 2A n A n 1 for all n 1. (A) 4950 (B) 4951 (C) 5050 (D) 5150 (E) After Taemin and Younghee had been painting a wall for five hours, Jaejoon helped them and they finished 3 hours later. If Jaejoon had not helped them, it would have taken Taemin and Younghee 5 more hours to complete the job. How long would it take in hours for Jaejoon to paint the wall alone? (A) 5 (B) 10 (C) 12 (D) 15 (E) For how many integers n between 100 and 200 is the greatest common divisor of 15 and n equal to 3? (A) 24 (B) 25 (C) 26 (D) 27 (E) 28 10

NMC Sample Problems: Grade 7

NMC Sample Problems: Grade 7. If Amy runs 4 4 mph miles in every 8 4. mph hour, what is her unit speed per hour? mph. mph 6 mph. At a stationary store in a state, a dozen of pencils originally sold for