2009 AMC 10B
1 、Each morning of her five-day workweek, Jane bought either a 50-cent muffin or a 75-cent bagel. Her total cost for the week was a whole number of dollars, How many bagels did she buy?
Solution
The only combination of five items with total cost a whole number of dollars is 3 muffins and bagels. The answer is .
2 、Which of the following is equal to ?
Solution
Multiplying the numerator and the denumerator by the same value does not change the value of the fraction. We can multiply both by , getting .
Alternately, we can directly compute that the numerator is , the denumerator is , and hence their ratio is .
3 、Paula the painter had just enough paint for identically sized rooms. Unfortunately, on the way to work, three cans of paint fell off her truck, so she had only enough paint for rooms. How many cans of paint did she u for the rooms?
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Solution
Losing three cans of paint corresponds to being able to paint five fewer rooms. So . The answer is .
4 、A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths and meters. What fraction of the yard is occupied by the flower beds?
Solution
Each triangle has leg length meters and area square meters. Thus the flower beds have a total area of 25 square meters. The entire yard has length 25 m and width 5 m, so its area is 125 square meters. The fraction of the yard occupied by the flower beds is . The answer is .
5 、Twenty percent less than 60 is one-third more than what number?
Solution
Twenty percent less than 60 is . One-third more than a number n is . Therefore and the number is 砂锅茄子. The answer is .
6 、Kiana has two older twin brothers. The product of their three ages is 128. What is the sum of their three ages?
Solution
The age of each person is a factor of . So the twins could be years of age and, conquently Kiana could be 128, 32, 8 or 2 years old, respectively. Becau Kiana is younger than her brothers, she must be 2 years old. So the sum of their ages is . The answer is .
7 、By inrting parenthes, it is possible to give the expression veral values. How many different values can be obtained?
Solution
The three operations can be performed on any of orders. However, if the addition is performed either first or last, then multiplying in either order produces the same result. So at most four distinct values can be obtained. It is easy to check that the values of the four expressions are in fact all distinct. So the answer is , which is choice .
8 、In a certain year the price of gasoline ro by during January, fell by during February, ro by 秋天的叶子during March, and fell by during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the nearest integer, what is
Solution
Let be the price at the beginning of January. The price at the end of March was Becau the price at the of April was , the price decread by during April, and the percent decrea was 小学古诗七步诗So to the nearest integer is . The answer is .
9 、Segment and interct at , as shown, , and . What is the degree measure of ?
成都天气15天Solution
is isosceles, hence .
The sum of internal angles of can now be expresd as , hence , and each of the other two angles is .
Now we know that .
Finally, is isosceles, hence each of the two remaining angles ( and ) is equal to .
10 花朵绘画、A flagpole is originally meters tall. A hurricane snaps the flagpole at a point meters above the ground so that the upper part, still attached to the stump, touches the ground meter away from the ba. What is ?
Solution
The broken flagpole forms a right triangle with legs and , and hypotenu . The Pythagorean theorem now states that 单针探底, hence , and .
(Note that the resulting triangle is the well-known right triangle, scaled by .)
11 、How many -digit palindromes (numbers that read the same backward as forward) can be formed using the digits , , , , , , ?
Solution
A ven-digit palindrome is a number of the form . Clearly, must be , as we have an odd number of fives. We are then left with . Each of the permutations of the t will give us one palindrome.
12、Distinct points , , , and lie on a line, with . Points and lie on a cond line, parallel to the first, with . A triangle with positive area has three of the six points as its vertices. How many possible values are there for the area of the triangle?
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