Exam 1: The Art of Problem Solving

Solve the problem. -A certain department store's profits for the months of July and August were a 6-digit palindrome, say $abc,def, satisfying these conditions: The store made less than $\$ 300,000$ in profits; a+b=(d+c)+1 b+c=8 $a \neq b$ ; and if a were decreased by 2 and b were increased by 3 , the sum of the two new numbers would be equal to the sum of $d$ and $e$ . By how much did the store surpass its 2-month goal of $\$ 245,856$ ?

Determine what the next equation would be, and verify that it is indeed a true statement. -2 × 4 = 3 × 5 - 7 4 × 6 = 5 × 7 - 11

Solve the problem. -A certain part of a metropolitan area has a palindromic, 5-digit area code, say abcde, satisfying these conditions: a, b and $c$ are all multiples of two and are all less than 8 ; $a \neq b$ but $a + b = 6$ ; $c \neq d$ but $a + c = 12$ ; $\mathrm { e } \neq \mathrm { b }$ or $\mathrm { d }$ ; $b + d \neq 4$ . Determine the area code.

Use your calculator to perform each calculation and observe the answers. Use inductive reasoning to determine which statement is true. - $\frac { ( - 3 ) \times 24 } { 8 \times 3 } ; \frac { ( - 14 ) \times 25 } { ( - 5 ) \times ( - 7 ) } ; \frac { - 3 \times ( - 40 ) } { ( - 2 ) \times ( - 12 ) \times ( - 7 ) } ; \frac { - 3.8 \times ( - 4 ) \times 1.3 \times ( - 5 ) } { ( - 9 ) \times ( - 9.4 ) \times ( 6.1 ) }$ What is true about expressions containing products of nonzero numbers in both the numerator and denominator?

The bar graph below shows the number of students by major in the College of Arts and Sciences. Answer the question. -Which majors, considered separately, include at least 15% of the total number of students in the College of Arts and Sciences?

Draw the next figure in the pattern.

Solve the problem using inductive reasoning. -How many line segments are determined by joining dots on the last two circles?

Use the indicated formula to find the sum. -Use $S = n ^ { 2 }$ to find the sum of $1 + 3 + 5 + \ldots + 499$ .

Determine the most probable next term in the sequence. -8, 13, 20, 29, 40, 53

Use problem solving strategies to solve the problem. -I'm thinking of a number between 4 and 10. If I square my number and find the difference of the digits, I will end up with the cube root of my number. What is my number?

Use the method of Gauss to find the sum. -1 + 2 + 3 + . . . + 650

Solve the problem. -Ethan was asked by his teacher to subtract 15 from a certain number and then divide the result by 5. Instead, he subtracted 5 and then divided the result by 15, giving an answer of 20. What would His answer have been if he had worked the problem correctly?

Use your calculator to perform the indicated operations. Give as many digits in your answer as shown on your calculator display. - $7 \cdot \pi$

Use problem solving strategies to solve the problem. -A mouse is at the bottom of a 10-foot-tall clock. Every hour he climbs up 3 feet. But when the clock strikes at the hour, he falls back 1 foot. If the mouse starts climbing at 8 a.m., at what time to the Nearest minute will it reach the top of the clock?

Use the indicated formula to find the sum. -Use $S = \frac { n ( n + 1 ) } { 2 }$ to find the sum of $1 + 2 + 3 + \ldots + 625$ .

Decide whether the argument is an example of inductive or deductive reasoning. -Fresh fruit costs more in winter. This is January. Therefore these fresh strawberries will cost more.

Use the method of successive differences to determine the next term in the sequence. -20, 31, 45, 62, 82, . . .

Decide whether the argument is an example of inductive or deductive reasoning. -His last four at bats were strikeouts. Therefore, the next will be a strikeout.

Determine the indicated term in the given sequence. -The 12th term of 3 , 9 , 27 , . . .

Use the indicated formula to find the sum. - $\text { Use } S = \frac { n ( n + 1 ) } { 2 } \text { to find the sum of } 1 + 2 + 3 + \ldots + 400 \text {. }$