Math Counts Test Review2010 Chapter

Let X be the cost of Sunday NewsX + (X + 0.95) = 7.25

2 X = 7.25 – 0.95 = 6.3

7. A copy of the Sunday Times costs $0.95 more than a copy of the Sunday News. Ms. Jones buys both papers for $7.25. How much does the Sunday News cost?

X = 3.15

Let A,B, C, D be the test scores. A B C D(B + C)/2 = 80, D – A = 12, C D & A B

Largest possible D = 80 + 12 = 92 with A = 80

11. Joshua has taken four tests. His median score is 80 points and the difference between his greatest score and his least score is 12 points. What is the maximum possible value of the mean of his scores?

Largest mean = (80*2+92+80)/4 = 40 + 23 + 20 = 83

Time in between chimes: 10/5 = 2 secondsAt 12 o’clock, there are 12 chimes

17. A grandfather clock chimes at the start of each hour. Beginning precisely at six o’clock the grandfather clock chimes 6 times with the final chime beginning exactly 10 seconds after six o’clock. If the chimes are uniformly spaced, how long after 12 o’clock does the twelfth chime begin?

The time before the 12th chime: 11*2 = 22

Total # of possible selection: 6 * 5# of desired outcome: 2 (r) x 4 (g) + 4(g) x 2(r)

18. A bag contains exactly six balls; two are red and four are green. Sam randomly selects one of the six balls and puts it on the table. Then he randomly selects one of the five remaining balls. What is the probability that the two selected balls are of different colors? Express your answer as a common fraction.

Probability = (8 + 8) / 30 = 16/30 = 8/15

Alternatively: P(r)(g) + P(g)(r) = 2/6 * 4/5 + 4/6 * 2/5 = 8/15

Note that EFG EABFG = 6 - 1 - 2 = 3, AB = 6

23. In rectangle ABCD, side AB measures 6 units and side BC measures 3 units, as shown. Points F and G are on side CD with segment DF measuring 1 unit and segment GC measuring 2 units, and lines AF and BG intersect at E. What is the area of triangle AEB?

Let H be the height of EFGH / (H + 3) = 3/6 = ½ 2H = H + 3 H = 3Area of EAB = 6 *( H + 3)/2 = 18

h(x) = 3x2 + 1 + 5x – 3 = 3x2 + 5x – 2 = 0 Let X1, X2 be the roots, by sum of the root,

27. If f(x) = 5x – 3, g(x) = 3x2 + 1 and h(x) = f(x) + g(x), what is the sum of the x-values for which h(x) = 0? Express your answer as a common fraction.

We have: X1 + X2 = -5/3

(X + Y) / 2 = 7 X + Y = 14 ----- (1)(X2 + Y2) /2 = 54 X2 + Y2 = 108 --- (2)

28. Two real numbers have an average of 7. The average of their squares is 54. What is the product of the two numbers?

From (1), (X + Y)2 = X2 + 2XY + Y2 = 196 --- (3)(3) – (2): 2 XY = 196 – 108 = 88Hence XY = 88/2 = 44

By AB = AC, ABC = ACBBy BD = BC, BDC = ACB

29. In triangle ABC, AB = AC and D is a point on AC so that BD bisects angle ABC. If BD = BC, what is the measure, in degrees, of angle A?

Hence ABC BCDWe get: CBD = ASince BD bisects ABC, A = ½ ABCWe have: 5 * A = 180 A = 180/5 = 36

A

B C

D

X + 43 = m * m ----- (1)X + 16 = n * n ------ (2)(1) – (2) m*m – n*n = 27(m + n) (m – n) = 27Since m, n (m > n) are integers, m+n & m – n are also positive integersPossible answers are: m+n = 27, m-n =1 -- (3) or m+n = 9, m-n =3 -- (4) From (3), m = 14, n = 13 mn = 14*13=182From (4), m = 6, n = 3 mn = 18, but wrong X

There are total of 37 scoresMedian is the 19th -- 78

G3. The distribution of the 37 test scores in a math class is given in the stem and leaf plot where 5|6 represents 56 points. What percent of the scores are at most 5 points from the value of the median? Express your answer to the nearest whole number.

There are 8 scores that arebetween 73 and 83:Answer: 8/37 = 0.216 = 22

Listing out the equations:a + b + c + d = 11 ------- (1)

G4. If a + b + c + d = 11, 2a + 3c = 19, b + 4d = 22, 4a + d = 14 and 5b + 3c = 5, what is the value of d?

2a + 3c = 19 ------- (2)b + 4d = 22 ------- (3)4a + d = 14 ------- (4)5b + 3c = 5 ------- (5)(2) + (3) + (4) + (5), we have:6a + 6b + 6c + 5d = 19+22+14+5 = 60 ------- (6)6 * (1) – (5): d = 66 – 60 = 6

Let Ca – cost of A, Cb – cost of B

G6. A shopper notes that a box of brand A cornflakes costs 50% more than a box of brand B and weighs 25% more than a box of brand B. According to this information, for equal weights, the cost of brand A is what percent more than the cost of brand B?

Let Wa – weight of A, Wb – weight of BUnit-Cost-A = Ca /Wa = (150% Cb)/(125% Wb ) = 6/5 Cb/Wb Difference in unit-cost: Ca /Wa - Cb/Wb = 6/5 Cb/Wb - Cb/Wb = 1/5 Cb/Wb

Percentage of the difference = (1/5 Cb/Wb )/ (Cb/Wb) = 1/5 = 20%

From 1/x + 1/y = 1/6, we get: (x+y)/xy = 1/66 x + 6 y = xy xy – 6x – 6y = 0

G8. How many ordered pairs of positive integers satisfy the equation, 1/X + 1/Y = 1/6 with x < y?

xy - 6x – 6y + 36 – 36 = 0 (x – 6) (y – 6) = 36 Since X, Y are positive integers, x-6, y-6 are integers

(x – 6) and (y – 6) are integer factors of 36 Since X < Y, we have (x-6) < (y-6)Possible values for (x-6): 1, 2, 3, 4, Answer: 4 (7, 42), (8, 24), (9, 18), (10, 15)

By 7 be the mean, median, 7 must be in the set

M5. A set of five positive integers has a mean, median and range of 7. How many distinct sets whose members are listed from least to greatest could have these properties? Note: {4, 6, 7, 7, 11} is one such set to include.

a

Let a, b, c, d, be differences of the other 4 numbers to 7, as shown.We have a + b = c + d, and a + d = 7We have a + b = c + d, and a + d = 7If a >= 5 d <=2 c + d <= 4 < a + d no fit answersIf a = 4, d = 3 c = 1, 2, 3 b = 0, 1, 2 3 answersIf a = 3, d = 4 b = 1, 2, 3 c = 0, 1, 2 3 answersIf a = 2, d = 5 a + b <= 2+2 = 4 < 5 no fit answersAnswer: 6

7

b c

d

200000 = 26 * 55

20! = 5*10*15*20 * 2*4*8*12*14*16*…

M7. What is the greatest common factor of 20! and 200,000?

Common factors: 26 * 54 = 200000/5 = 40000

Start with 100 pennies - $1.00Need to remove 51 coins

M8. Harry has 49 coins totaling $1.00. Harry has only pennies, nickels, dimes and/or quarters. What is the smallest number of dimes he could have?

Every replace of 1 dime removes 9 coinsEvery replace of 1 nickel removes 4 coinsEvery replace of 1 quarter removes 24 coinsWe can’t remove 51 coins with zero or even number of dimes as nickels and quarters remove even # of coinsIf use 1 dime, we need to remove 51 – 9 = 42 coins, which can’t be done with multiple 4 & 24 If use 3 dimes, we need to remove 51 – 27 = 24 coins – can be achieved with 1 quarter OR 6 nickelsAnswer: 3

There are total of 7 balls that are not of color yellow.

M9. A bag contains 3 red balls, 4 green balls, and 5 yellow balls. If balls are drawn one at a time without replacement, what is the probability that the first yellow ball is drawn on the eighth draw? Express your answer as a common fraction.

Prob. = 7/12 * 6/11 * 5/10 * 4/9 * 3/8 * 2/7 * 1/6 * 1

= 7/12 * 6/11 * 5/10 * 4/9 * 3/8 * 2/7 * 1/6 * 1

= 7/12 * 6/11 * 5/10 * 4/9 * 3/8 * 2/7 * 1/6 * 1 = 1/3 * 1/11 * 1/3 * 1/8

= 1/792