Warm-Up A palindrome is a number that reads the same forward as backward. How many 3-digit palindromes are multiples of 3?

Transcription

1 Warm-Up 7 1. A triangle with a height of 24 inches has the same area as a rectangle 12 inches by 6 inches. How many inches long is the base of the triangle that corresponds to the 24-inch height? 2. Rohan plans to open a bank for his friends. Some friends will allow him to keep their money in savings, and others will borrow money from him. The charts below describe the amounts that students are saving and borrowing from him. What is the least possible value, in dollars, of the money that Rohan is currently keeping in his bank? Express your answer to the nearest whole number. Amount put in for savings Value of accounts Number of accounts $0.01-$ $20.01-$ $40.01-$ $60.01-$ Amount taken for loans Value of loan Number of loans $0.01-$ $20.01-$ $40.01-$ $60.01-$ A group of N students, where N < 50, is on a field trip. If their teacher puts them in groups of 8, the last group has 5 students. If their teacher instead puts them in groups of 6, the last group has 3 students. What is the sum of all possible values of N? 4. An environmental agency needs to hire a number of new employees so that 85 of the new employees will be able to monitor water pollution, 73 of the new employees will be able to monitor air pollution and 27 of the new employees will be able to monitor both. What is the minimum number of employees that need to be hired? 5. When the length of an object is given to the nearest quarter-inch, the measurement could be off by a maximum of what part of an inch? Express your answer as a common fraction. 6. What is the sum of all integer values of x such that [ is an integer? 7. A palindrome is a number that reads the same forward as backward. How many 3-digit palindromes are multiples of 3? 8. The 52-digit whole number N = consists of fifty zeros between the digits 2 and 3. How many digits are zero in the decimal expansion of 1? 9. The product of the base seven numbers 24 7 and 30 7 is expressed in base seven. What is the base seven sum of the digits of this product? 10. Let a, b, c, d and e be whole numbers. The sum of the four numbers on each of the five longest line segments of the star is the same. If a = 18, what is the value of b? MATHCOUNTS

2 1 Warm-Up & Solutions 1. h=24 Area rec tangle = 12 6 = 72 base Area triangle = 72 = 1 base height = 1 base Therefore: base = 144 = Key is to understand: the least possible value. This means having the lowest savings, (deposits) and the highest loans. A table helps to organize the data: Savings (deposits or input) Loans (output) Range Number Minimum Amount Range Number Maximum Amount $ $ $ $ $ $ $ $ Total $ Total $ Difference: $ $360.00=$80.15 or $80

3 2 3. Facts: Number of students, N, < 50. Grouped by 8 s there are 5 left over, grouped by 6 s there are 3 leftover. N + 5 and N + 3 are the groups 8 6 Make two lists: N for Groups of 8 N for Groups of and 21 are the only two common solutions. They asked for the sum of all possible: = This find of problem is best solved using a Venn diagram: WP=85, AP=73, and there are 27 for both: WP AP WP only = = 58 AP only = = Minimum = =

4 3 5. Length is given to nearest ¼ of an inch. What is the maximum error or the amount that it could be off? 1/4 3/4 1/2 INCH To either side, the next increment would be used. Right in the middle, the error is ½ of the increment or 1/8 of an inch. 1/8 6. The sum of all integer values of X that satisfy: 67 Key is that 67 is a prime so its factors are 67 2x 23 and 1. Because it is not limited to positive integers, the results can be negative as well, +/-67 ans +/-1: 2x 23= 67 2x= 90 x= 45 2x 23= 1 2x= 24 x= 12 2x 23 = 67 2x= 44 x= 22 2x 23= 1 2x= 22 x= ( 22) + 11=

5 4 7. Palindrome: Number that reads the same forward and backward, example is 101. Need to find all 3 digit palindromes that are multiples of 3. To be a multiple of three, the sum of the digits must add to a multiple of three: Strategy, list all palindromes and then select those that are a multiple of 3: 101, 111, 121, 131, 141, 151, 161, 171, 181, 191: 3 for 100 s Continue this for 200 through 900 and we get 30 numbers: A 52-digit whole number, N, has 50 0 s between the 2 and 3: How many zeros in N 2? Turn it into a simpler case with just two zeros: 2003 times 2003 and we find the answer Notice that the two zeros repeats: 2009 and there is another 0 in 401. What happens if the numbers are 20003: The zeros repeat and there is another term with 1 less zero. Therefore for 50 zeros there will be or 99 zeros: 99