NMC Sample Problems: Grade 7

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1 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 dollars is marked 0% off. Assume that the state s sales tax rate is 8%. If you purchase one dozen of the pencils at the store at the sale price, how much would you need to spend for pencils including the sales tax? $.0 $.06 $.07 $.08 $.09. The area of a rectangle is 08 cm, where its length is cm longer than its width. Find the perimeter of the rectangle. 0 cm 4 cm 8 cm 4 cm 46 cm 4. What is the product of all valid solutions of the fractional equation 9 0 x 4 x = (x )? None of these. Find the value of If three numbers are randomly selected without replacement from 0,,,, and 4, how many three digit numbers can we make? If Brittany gets a 97 on her next math test, her average will be 89. However, if she gets 7, her average will be 84. How many tests has Brittany already taken? of 9

2 8. We partition the rectangle ABCD into seven congruent rectangles like the figure below. If the area of the ABCD is 84 cm, what is the perimeter of a small rectangle in cm? (Note: Figure not drawn to scale!) A B D C Tom plants white rosebushes and red rosebushes in a row. How many different ways can he plant at least four white bushes consecutively? Two circles, A and B, overlap each other as shown below. The area of the common part is of the area of Circle A, and of the area of Circle B. What is the ratio of the radius of Circle A to 8 that of Circle B? (Note: Figure not drawn to scale!) A B : : 4 : : 4 6 :. 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? Increasing a number by.% is equivalent to multiplying it by which of the following? The greatest common divisor of two possible numbers is 4 and the least common multiple is 8. Find the sum of the two numbers of 9

3 4. Martha s parents presented her a saving account with the balance of $8,000 on her birthday this year. When she was born, her parents deposited $,000, which tripled every four years. How old is Martha? 4 years old 8 years old years old 6 years old 7 years old. Let A and B be natural numbers such that A = B. Let M be the least common multiple of A and B, and let D be the greatest common divisor of A and B. If the product of M and D is 6, then find B A A town has a basketball team and a football team, and both teams play once a week. The basketball team wins out of games, and the football team loses out of games. What is the probability that both teams win on the same week? Solve the equation x [ (x + )] = Jamie buys a 9 inch long sub sandwich to share with his brothers. He cuts it into three pieces so that the mid-sized piece is inch longer than the shortest piece and the shortest piece is 9 inch shorter than the longest piece. How long in inches is the mid-sized piece? In a middle school, there are 4 math classes. The average math scores of the classes are 79, 8, 84, and 7, and the number of students in each class is 0,,, and 6, respectively. Find the average score of the 4 math classes combined Let x + y = and x + y =. Find the value of x + y When the base of a triangle increases by 0% and the height to the base decreases by 0%, the area of the triangle will. increase by 0% decreased by 0% increase by % increase by 8% decrease by 8%. Let a b = ab a + b for any integers a and b. Find the value of ( ( )) of 9

4 . Find the area bounded by y = x + and the x-axis between x = and x = Use the list of numbers below to answer the question that follows. 8,, 7,, k,, 9, 0 If the mean of the list of numbers above is, what is the median of the list? None of these. Which of the following is equivalent to the below expression? ( ) x 4 ( ) y y x 6 x 0 y 6 y 9 x 8 y x 4 y x None of these 6. 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? Which of the following number is the smallest when p < and 0 < q <? p q p p q 8. Amy and Bryan walk toward each other from A to B. Amy walks at 0 feet per minute and Bryan takes minutes to walk a mile. The distance between A and B is miles. If they start walking to each other at the same time, approximately how long will it take for them to meet? 40 minutes hour 4 minutes hour hours hour 0 minutes 9. Evaluate ,00,00 499,999 00,999 00,00 499,00 0. 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? 7:00 PM 8:00 PM 9:00 PM 8:0 PM 9:0 PM 4 of 9

5 . If x and y are distinct prime numbers, which of the following numbers must be odd? xy (x + ) y + x + y + xy x + y. 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? α β δ γ α + β + γ + δ = 80 γ + δ = 90 α + β = 90 β + δ = 90 α + γ = β + δ. The Fine Art Center charged an admission fee of $ per adult and $ per child. On April, the center earned $0 in total. On the next day, the Center earned $6 and had the number of adults decreased by % and the number of children increased by 6%. How many children did the Center have on April? I had. gallons of 00% pure grape juice. After using 0. gallons of juice, I mixed 0. gallons of water with the remaining juice. Then I drank 0. gallons of the juice and mixed 0. gallons of water with the remaining juice. How pure is the remaining juice as a percentage? 4% 64% 7% 80% 9%. In the table below, find the number in the th row and the 8 th column There is a clock (which has two hands) in my classroom. After Mathematics class, I checked the time. The clock in the classroom indicated that is was :6 pm. If the hour hand moves continuously between and 6, find the measure of the acute angle between two hands of 9

6 7. An urn contains red and black balls. Tom first withdraws a ball from the urn and Mary withdraws a ball from the urn without replacement. They keep withdrawing a ball alternately until a red ball is selected. Find the probability that Tom selects the red ball A lattice point is an ordered pair of integers (x, y). How many lattice points (including the end points) are there on the line segment joining the points (4, 0) and (8, 7)? Brian and Emma share the labor to build a hut. It would take 0 days to complete the hut if Brian works the first 6 days and then Emma works the rest 4 days after Brian. However, it would take days if Brian works the first 9 days and then Emma works the rest days after Brian. If Brian works alone, how many days will it take to complete the hut? 0 days days days days 4 days 40. Tom read a book in days. On the first day he read / of the book and pages more. The next day he read /0 of what remained and 0 pages more. On the third day he read /4 of what remained and 0 pages more. How many pages were there in the book? Sumin rolls three distinct dice and gets a, b, and c. Find the probability of a + b + c = Find the number of the ordered pairs (a, b) such that the four-digit number, a6b, is a multiple of Two points P and Q lie on a sphere with a radius of 0. The length of the line segment joining P and Q is 0. What is the shortest distance between the segment P Q and the center of the sphere? The population in town A declined at a constant rate from 0,000 in the year 990 to 9,0 in the year 998. The population in town B increased at a constant rate from,000 in the year 994 to,640 in the year 998. If the rates of change of the population in the two towns remain the same, in approximately what year will the populations in the two towns be equal? of 9

7 4. Suppose that f(x) = ax + b, where a and b are real numbers. Given that f(f(f(x))) = 7x + 9, what is the value of a + b? Suppose that a flight from Atlanta to Los Angeles takes hours, while a flight from Los Angeles to Atlanta takes hours. The difference in time is due to a wind blowing from west to east. How long, in minutes, would it take to fly between the two cities if there were no wind? The width of a rectangle is % of the length. The perimeter of the rectangle is 08 cm. Find the width of the rectangle. 7 cm 4 cm 0 cm 40 cm 44 cm 48. A water tank had some water, and gallons of water was added to the tank each day starting from March st. If there are 0 gallons of water in the tank at the end of April th, how many gallons were in the tank before March st? The ratio of the interior angles of a quadrilateral is : : : 6. Find the largest angle of this quadrilateral If x < 0, then which of the following must be positive? x x x x + x x x x. The average Mathematics test score of all 7th grade students is 76. The average score of all the boys is 7 and the average of all the girls is Find the ratio of the number of boys to the number of girls as a fraction Evaluate (a b) when a = and b = The sum of two numbers is. Eight more than three times the second is equal to four times the first. Find the two numbers. and 0 and 9 and 4 8 and 7 and 6 7 of 9

8 4. There are red balls and 7 blue balls in Bowl A, and 8 red balls and blue balls in Bowl B. Mandy rolls a dice and if a prime number is up, Bowl A is selected; otherwise Bowl B is selected. Then Katie draws a ball randomly from the selected bowl. Find the probability of Bowl A being selected given that a red ball is drawn.. Let f(x) + f( x) = 6 for all x. Find f( 00 ) + f( ) + + f( 00 ). 6. A circle is inscribed in the triangle ABC whose side lengths are 8 cm, 0 cm, and 4 cm. Find the radius of the circle in cm. 7. Find the smallest three-digit integer whose remainder is when it is divided by,, and Consider a sequence of numbers, A, A, A,. The entries of the sequence satisfy the following properties. Find A. A = and A = A n+ = A n+ A n for all n > 0 9. Evalaute: Evalaute: of 9

9 KEYS [] [6] [] [46] [] [7] [] [47] [] [8] [] [48] [4] [9] [4] [49] [] [0] [] [0] [6] [] [6] [] [7] [] [7] [] [8] [] [8] [] [9] [4] [9] [4] [0] [] [40] [] 97 [] [6] [4] [6] 6 [] [7] [4] [7] 09 [] [8] [4] [8] 048 [4] [9] [44] [9] 749 [] [0] [4] [60] 00

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