MATH Midterm
Use the lattice algorithm to perform each of the following. a. 1798+7643 b. 7859+6576 ( Go to note card to check answer)
a. 9441 b. 14435
Ben claims that zero is the same as nothing. Explain how you as a teacher would respond to Ben's statement.
Ben is incorrect. Zero is used as a place holder in the Hindu-Arabic system. It is used to differentiate between numbers like 11 and 1001
A car trip took 3 hours at an average speed of 45 mph. Mentally compute the total number of miles traveled.
To mentally compute the product, first separate the tens and the units of the speed, which is also the larger number. Part 2 45=40+5 Part 3 Multiply the number of hours by both the tens and the units of the speed. 3×40=120 Part 4 3×5=15 Part 5 Add the two products. 120+15=135 Part 6 The trip was approximately 135 miles.
In a fraternity with 41 members, 18 take mathematics, 5 take both mathematics and physics, and 8 take neither mathematics nor physics. How many take physics but not mathematics?
Construct a Venn diagram to help solve the problem. math | | physics Part 2 Since 5 took both math and physics, a 5 has been placed in the intersection of those two circles. math | 5 | physics Part 3 The problem states that 18 took math. 13 took only math. Part 4 This has been entered into the diagram. math - 13 | 5 | physics Part 5 The problem states that 8 take neither math nor physics. This value goes into the region which includes neither circle. math - 13 | 5 | physics. 8 (outside of circle) Part 6 Looking at the diagram, and counting, 26 are accounted for. Part 7 There are 41 members of the fraternity, and 26 are accounted for. 15 must have taken physics, but not math. Part 8 The correct diagram is below. math - 13 | 5 | 15 - physics. 8 (outside of circle)
Find the sum 56+57+58+59+...+160+161
Explanation: A good strategy to determine the sum is to pair together entries so that the sum of each pair is the same. Part 2 Start by finding the sum of the first and last terms. 56+161=217 Part 3 Note that not only do the first and last terms have a sum of 217, but so do the second and next-to-last terms. 57+160=217 Part 4 This pattern continues with the third and third-to-last terms, and so forth. Let S=56+57+58+59+...+160+161. If you regroup S as (56+161)+(57+160)+..., then S=217+217+...+217. Part 5 217+217+...+217 is equal to the product of 217 and the number of pairs in the original sum. If there were 10 terms in the original sum, then there would be 5 pairs with a sum of 217. Therefore, S=217+217+...+217 would equal 217•5. Note that this method only works for sums with an even number of terms. Part 6 Notice that the first 4 terms are 56, 57, 58, and 59. Subtracting the first term from the fourth term gives a difference of 3, so add one to the difference between the last number in the sum and the first number in the sum to find the number of terms. There are 106 terms in the original sum, 56+57+58+59+...+160+161 because 161−56+1=106 Part 7 Therefore, there are 1062=53 pairs with a sum of 217. Multiply to find the sum S. S=53•217=11,501 Part 8 The sum 56+57+58+59+...+160+161 is 11,501.
For each of the following, find whole numbers to make the statement true, if possible. a. 70 ÷ 7 = ____ b. _____ ÷ 21 = 1 c. 35 ÷ ___ = 35
For any whole numbers a and b, with b≠0, a÷b=c if and only if c is the unique whole number such that b•c = a. Part 2 a. To solve this equation, carry out the calculation. Determine which operation to perform in order to find the missing number. Since the missing value is alone on the right side, the given operation of division will be performed to determine the missing value. Part 3 To solve the equation, divide 70 by 7. 70/7=10 Part 4 Thus, 70÷7=10. Part 5 b. To solve this equation, carry out the calculation. Determine which operation to perform in order to find the missing number. Given a÷b=c if and only if b•c = a, multiplication must be performed to determine the missing value which corresponds to a. Part 6 Since the missing value must equal 21•1, multiply 21 and 1 to determine what the missing value must be. 21•1=21 Part 7 Thus, 21÷21=1. Part 8 c. To solve this equation, carry out the calculation. Determine which operation to perform in order to find the missing number. Given a÷b=c if and only if b•c = a, where the missing number corresponds to b, division must be performed between the two known numbers such that a÷c=b. Part 9 Since the missing value multiplied by 35 must equal 35, divide 35 by 35 to determine what the missing value must be. 35/ 35=1 Part 10 Thus, 35÷1=35.
Simplify each of the multiplications in parts (a) through (d) below using properties of exponents. Use the given value as the base.
For whole numbers a and b and natural numbers m and n, the properties below are true. am•an=am+n a^m x a^n=a^m+n a^n•bn=(ab)n Part 2 a. 79•74 (use 7 as the base) Use the property am•an=am+n to simplify this expression in the most efficient way. Part 3 Rewrite 79•74 using the property am•an=am+n. 79•74=79+4=713 Part 4 b. 25•27•210 (use 2 as the base) Use the property am•an=am+n to simplify this expression in the most efficient way. Part 5 Rewrite 25•27•210 using repeated applications of the property am•an=am+n. 25•27•210=25+7+10=222 Part 6 c. 1412•212•712 (use 14 as the base) Use the properties an•bn=(ab)n and am•an=am+n to simplify this expression in the most efficient way. Part 7 Rewrite 212•712 as a single exponential expression using the property an•bn=(ab)n. 212•712=(2•7)12=1412 Part 8 Now rewrite the expression 1412•1412 using the property am•an=am+n. 1412•1412=1424 Part 9 Therefore, 1412•212•712=1424. Part 10 d. 45•87•1283 (use 2 as the base) Use the properties amn=amn and am•an=am+n to simplify this expression in the most efficient way. Part 11 Rewrite 4 as an exponential expression. 4=22 Part 12 This means 45=225. Simplify 225 using the property amn=amn. 225=210 Part 13 So 45=210. Rewrite 87 as an exponential expression with a base of 2 using the property amn=amn in a similar manner. 87=237=221 Part 14 Rewrite 1283 as an exponential expression with a base of 2 using the property amn=amn in a similar manner. 1283=273=221 Part 15 Thus, 45•87•1283=210•221•221. Rewrite 210•221•221 using repeated applications of the property am•an=am+n. 210•221•221=252 Part 16 Therefore, 45•87•1283=252.
Using a calculator, Ralph multiplied by 10 when he should have divided by 10. The display read 500. What should the correct display be?
If Ralph multiplied by 10 when he should have divided, the number is 10×10, or 100, times too large. Part 2 Thus you need to divide 500 by 100. 1005500 Part 3 The correct display should be 5.
Which of the sets can be placed in a one-to-one correspondence? a. {1,2,3,4} and {m,n,f,g} b. {a,b,c,d,...,u} and {1,2,3,...,21} c. {a|a is a letter in the word eggs} and {1,2,3,4}
If the elements of sets P and S can be paired so that for each element of P there is exactly one element of S and for each element of S there is exactly one element of P, then the two sets P and S are said to be in one-to-one correspondence. Part 2 a. Since {1,2,3,4} and {m,n,f,g} have 4 elements, then they can be placed in a one-to-one correspondence. Part 3 b. List the sets out in full. The first set is {a,b,c,d,...,u} and has 21 elements. Part 4 The second set is {1,2,3,...,21} and has 21 elements. Part 5 Since these numbers are equal, the sets can be placed in a one-to-one correspondence. Part 6 c. The set {a|a is a letter in the word eggs} has three elements. Part 7 The set {1,2,3,4} has four elements. Therefore, they cannot be placed in a one-to-one correspondence.
Shown to the right is a magic square (all rows, columns, and diagonals sum to the same number). Find the value of each letter. 9 - a- 19 28 -18- b c - d - 27
In a magic square all rows, columns, and diagonals have the same sum. The sum can be found in the given magic square because the diagonal which falls from the top left to the bottom right is filled. Part 2 The sum to use can be found from the given diagonal. The sum is 9+18+27=54. Part 3 So, each of the rows, columns, and diagonals must have a sum of 54. Use this sum to find the missing numbers. Start with the far right column, where only one number is missing. The sum of the column must be 54. Find the missing number, b. 19+b+27 =54 b=54−27−19 b=8 Part 4 Now find the missing number in the top row, a=54−9−19=26 9 - a - 19 28-18-8 c- d - 27 Part 5 Using the number just found for the top row, find the missing number in the middle column. d=54−26−18=10 9 - 26- 19 28 - 18- 8 c - d - 27 Part 6 Finally, find the missing number in the bottom row. c=54−10−27=17 9 - 26- 19 28- 18- 8 c - 10-27 Part 7 The completed magic square is shown on the right. Notice that each row, column, and diagonal has a sum of 54. 9-26-19 28-18-8 17-10-27
Cookies are sold singly or in packages of 5 or 15. With this packaging, how many ways can you buy 30 cookies?
In order to find the number of possible ways to buy 30 cookies, write out the ways 30 cookies can be purchased in packages of 1, 5, and 15 Part 2 Create a table that represents the number of packages purchased, where each row represents a different combination of packages. Part 3 Several different possible ways to buy 30 cookies are shown in the following table. single cookies - packs of 15 - packs of 5 30 - 0 - 0 25 - 0 - 1 20 - 0 - 2 15 - 1 - 0 Part 4 The complete table is shown below. single cookies - packs of 15 - packs of 5 30 - 0 -0 25 - 0 - 1 20 - 0 - 2 15 -1 -0 15 - 0 -3 10 - 1 - 1 10 - 0 - 4 5 - 1 - 2 5 - 0 - 5 0 - 2 -0 0 - 1 - 3 0 - 0 - 6 Part 5 The table lists all possible combinations in which 30 cookies can be bought. Part 6 Since each line of the table is different, the total number of lines equals the number of possible combinations. Part 7 The number of possible combinations is 12.
Question content area top Part 1 John claims that he can get the same answer to the problem below by adding up (begin with 6+7) or by adding down (begin with 9+7). He wants to know why and if this works all the time. How do you respond? 9 7 + 6 ------
It does work all of the time. This is because 9,7, and 6 are whole numbers, so the commutative and associative properties hold and allow the three numbers to be added regardless of their order.
Question content area top Part 1 John claims that he can get the same answer to the problem below by adding up (begin with 6+7) or by adding down (begin with 9+7). He wants to know why and if this works all the time. How do you respond? 9 7 + 6
It does work all of the time. This is because 9,7, and 6 are whole numbers, so the commutative and associative properties hold and allow the three numbers to be added regardless of their order.
Kathy stood on the middle rung of a ladder. She climbed up 6 rungs, moved down 2 rungs, and then climbed up 7 rungs. Then she climbed up the remaining 1 rung to the top of the ladder. How many rungs are there in the whole ladder?
Start by drawing a sketch of the ladder. Since a ladder with an even number of rungs does not have a middle rung, if Kathy is on the middle rung, then the ladder must have an odd number of rungs. middle rung Part 2 Next, notice that there are the same number of rungs above the middle rung as there are below the middle rung. If Kathy were to climb 3 steps to reach the top of the ladder, then there would be 3 rungs above the middle, 3 rungs below the middle, and the middle rung, for a total of 7 rungs. Part 3 The given ladder, however, has considerably more than 7 rungs. To determine the number of rungs on the given ladder, first determine the number of rungs above the middle rung. Since Kathy begins on the middle rung and ends on the top rung, her net change in rungs will give the number of rungs above the middle. Part 4 Represent the number of rungs Kathy moved up as positive integers, and the number of rungs she moved down as negative integers. Kathy climbed up 6 rungs. Represent that with positive 6. She moved down 2 rungs. Represent that with −2. She then went back up 7 rungs. Represent that with positive 7. Finally, she climbed the 1 remaining rung to get to the top rung. Represent that with positive 1. Part 5 The net change in rungs is 6+(−2)+7+1=12. Part 6 So, there are 12 rungs above the middle rung. Therefore, there are 12 rungs below the middle rung, and there is the middle rung. Determine the total number of rungs on the ladder. Part 7 There are 12+12+1=25 rungs on the ladder.
A popular brand of pen is available in 24 colors and 6 writing tips. How many different choices of pens do you have with this brand?
The Fundamental Counting Principle If you can choose one item from a group of M items and a second item from a group of N items, then the total number of two-item choices is M•N. Part 2 The number of ways that you can choose the pens is found by multiplying the number of colors (M) by the number of writing tips (N). M•N=24•6 Part 3 Thus, the answer is obtained by evaluating 24•6. There are 144 different choices of pens with this brand.
Round to the nearest hundred. 3672 and 4237
To round to a certain place, you first locate the digit for that place. In this problem, you want to round to the nearest hundred. Part 2 In 3672, the digit 6 is in the hundreds place. Part 3 The next step in rounding is to look at the digit to the right of the hundreds place. Part 4 The digit in the tens place, 7, is to the right of 6. Part 5 7 is higher (greater) than 5. Part 6 Since 7 is 5 or higher, you want to round up. To round up, change the 6 to 7 and change all digits to the right to zero. The thousands digit, 3, remains the same. 3672 rounded to the nearest hundred is 3700. Part 7 Now suppose you want to round 4237 to the nearest hundred. Identify 2 as the digit in the hundreds place and 3 as the digit to the right. Part 8 Because 3 is less than 5, you want to round down. To round down, keep the hundreds digit, 2, as 2 hundreds and change the digits to the right,3 and 7, to zeros. The thousands digit, 4, remains the same. 4237 rounded to the nearest hundred is 4200.
If n(A)=37, n(B)=42, and n(A ∩ B)=11 what is known about n(A ∪ B)?
To solve this problem, use the formula for n(A ∪ B), shown below. n(A ∪ B)=n(A)+n(B)−n(A ∩ B) Part 2 The follow information is given. n(A)=37 n(B)=42 n(A ∩ B)=11 Part 3 Use the information from above to find n(A ∪ B). n(A ∪ B) = n(A)+n(B)−n(A ∩ B) = 37+42−11 = 68 Part 4 Therefore, n(A ∪ B)= 68
Cats cannot be dogs. Use the two sets below to draw a Venn diagram depicting their relationship. cats and dogs
When drawing a Venn diagram, the universal set is represented by a large rectangle, and the other sets are indicated by circles inside the rectangle. If two sets share elements, their circles will overlap. If a set is a proper subset of another set then the subset's circle will be entirely within the circle for the other set. Carefully examine the Venn diagrams to choose the one that depicts the situation correctly. Part 2 Let C=cats and D=dogs. First, find the relationship between the two sets. Part 3 There are four ways sets can be related, and each can be represented by a Venn diagram. One way is that D may be a subset of C, meaning that all members of D are also members of C. Another is that C may be a subset of D, meaning that all members of C are also members of D. Another is that C and D may be disjoint, meaning that the two sets have no members in common. And one last way is that C and D may be overlapping sets, meaning that the two sets share some of the same members. Part 4 Find the relationship between the two sets. The sets C and D are disjoint because there are no cats that are dogs. Part 5 Draw a Venn diagram with two circles showing the given relationship between two sets. The correct Venn diagram is shown to the right. C (separate bubbles) D
Answer parts a and b. a. If n(A ∪ B)=32, n(A ∩ B)=10, and n(B)=11, find n(A). b. If n(A)=27, n(B)=17, and n(A ∩ B)=12, find n(A ∪ B).
a. If n(A ∪ B)=32, n(A ∩ B)=10, and n(B)=11, find n(A). The sum of the elements of two sets minus the elements in their intersection will give the number of elements in their union. This translates to the following formula. n(A ∪ B)=n(A)+n(B)−n(A ∩ B) Part 2 There are 32 elements in (A ∪ B). Part 3 There are 11 elements in B. Part 4 There are 10 elements in (A ∩ B). Part 5 Substitute the known values into the formula to determine the value of n(A). n(A ∪ B) = n(A)+n(B)−n(A ∩ B) This is the formula. 32= n(A)+11−10 Substitute. 32=n(A)+1 Subtract on the right side. Part 6 n(A)= 31 Solve for n(A). Part 7 Therefore, if there are 31 elements in A and 11 elements in B, and the intersection of the two sets is 10 elements, then the union of the two sets will contain 32 elements. Draw a diagram to check the answer. Part 8 b. If n(A)=27, n(B)=17, and n(A ∩ B)=12, find n(A ∪ B). Use the formula for n(A ∪ B). n(A ∪ B)=n(A)+n(B)−n(A ∩ B) Part 9 There are 27 elements in A. Part 10 There are 17 elements in B. Part 11 There are 12 elements in (A ∩ B). Part 12 Substitute the known values into the formula from above to determine the value of n(A ∪ B). n(A ∪ B) = n(A)+n(B)−n(A ∩ B) This is the formula. =27+17−12 Substitute. =32 Simplify. Part 13 Therefore, if there are 27 elements in A and 17 elements in B, and the intersection of the two sets is 12 elements, then the union of the two sets will contain 32 elements. Draw a diagram to check the answer.
a. P is equal to the set containing r, q, p, and d. b. The set consisting of the elements 5 and 1 is a proper subset of {5,1,8,12}. c. The set consisting of the elements 0 and 4 is not a subset of {4,5,2,3}.
a. P is the set whose elements are r, q, p, and d. The set that contains these elements is written as {r,q,p,d}. Part 2 Therefore, the correct expression is P={r,q,p,d}. Part 3 b. The symbol ⊂ is the symbol for "is a proper subset of." Therefore, the correct expression is {5,1}⊂{5,1,8,12}. Part 4 c. The symbol ⊈ is the symbol for "is not a subset of." Therefore, the correct expression is {0,4}⊈{4,5,2,3}.
For each of the following base ten numerals, give the place value of the underlined digit. a. 276,384 b. 1,927,000
a. Recall that place value in the base ten numeration system is based on powers of ten. The place value of each digit represents a power of ten. Part 2 The power of 10 represented by the place value of the underlined digit in 276,384 is 103, or 1,000. Part 3 Thus, the place value of 6 in 276,384 is thousands. Part 4 b. The power of 10 represented by the place value of the underlined digit in 1,927,000 is 106, or 1,000,000. Part 5 Thus, the place value of 1 in 1,927,000 is millions.
Let A={3,4,5,6,7}. a. How many subsets does A have? b. How many proper subsets does A have?
a. The number of distinct subsets of a finite set A is 2n, where n is the number of elements in set A. Part 2 There are 5 elements in A. Part 3 Substitute 5 for n and evaluate the expression. Because 25=32, there are 32 subsets of A. Part 4 b. Set A is a proper subset of set B, symbolized by A ⊂ B, if and only if all the elements of set A are elements of set B and A≠B (that is, set B must contain at least one element not in set A). Note that no set is a proper subset of itself. Part 5 There is only one subset of A that is not a proper subset, and that subset is A itself. Therefore, for a set with n elements, the number of proper subsets of the set is 2n−1. Part 6 Substitute 5 for n and evaluate the expression. Note that 25−1=31. Thus, there are 31 proper subsets of A.
For each of the following find, if possible, a whole number that makes the equation true. a. 3•Z=18 b. 54=36+3•Z c. Z•(8+6)=Z•8+Z•6
a. This equation says three times a number is 18. To solve for Z, divide both sides by 3 because division is the inverse of multiplication. 3•Z/3 = 18/3 Z = 6 Part 2 b. The first step to solve for Z is to subtract 36 from both sides. 54−36 = 36+3•Z−36 18 = 3•Z Part 3 Divide both sides by 3. 18/3 = 3•Z/3 6 = Z Part 4 c. Use the distributive property of multiplication on the left side of the equation to solve for Z. Z•8+Z•6=Z•8+Z•6 Part 5 Since the left side and the right side are the same for any value of Z, the solution is any whole number. Every whole number, when substituted for Z, results in a true statement.
Use the following pattern to complete parts (a) and (b) below. 13 + 39 = 13•2^2 13+39+65 =13•3^2 13+39+65+91=13•4^2
a. What is an inductive generalization based on this pattern, where n is equal to the number of terms being added? To find an inductive generalization based on this pattern, examine the pattern and derive a formula to predict the sum of the terms in terms of n. 13 + 39 = 13•2^2 13+39+65 =13•3^2 13+39+65+91=13•4^2 Part 2 First, notice that each term has a common factor. The common factor is 13. Part 3 Divide the common factor out of the entire sequence. 13/13+ 39/13+ 65/13+ 91/13 =13•4^2/ 13 1 + 3 + 5 + 7 = 42 Part 4 The variable n represents the number of terms being added. The sequence in the previous step has 4 terms being added. Thus, n2=42. Part 5 Remember, the common factor is 13. The formula that predicts the sum of the terms is 13n2. Part 6 b. Based on the generalization in (a), find the sum of the sequence 13+39+65+91+...+169. To obtain the sum of the terms, first identify which term in the pattern is 169. 13+39+65+91+...+169 Part 7 Find a formula for the nth term of the sequence. To simplify this process, first divide each term by 13. 13,39,65,91→13/13,39/13,65/13, 91/13→ 1,3,5,7 Part 8 Find the nth term. Term Number -Term 1 -1 2 -3 3 - 5 4 -7 • - • • - • • - • n - 2n−1 Part 9 Since the sequence was divided by 13 earlier, multiply the formula for n by 13. (2n−1)•13 = 26n−13 Part 10 Thus, the nth term of the sequence 13, 39, 65, 91,... is equal to 26n−13. Part 11 Set this formula equal to the given value, 169, and solve for n. 26n−13 =169 n = 7 Part 12 Use the formula found in part (a), sum=13n2, to calculate the sum of the first seven terms. sum = 13n2 = 13(7)2 = 637 Part 13 Thus, the sum of the first seven terms is 637.
Find the sum of the sequence. 53+54+55+56+...+144
here is a good strategy for finding the sum of a sequence such as this one. Start by finding the sum of the first and last terms. 53+144=197 Part 2 Note that not only do the first and last terms sum to 197, but so do the second and next-to-last terms. 54+143=197 Part 3 This pattern continues with the third and third-to-last term, and so forth. Let s=53+54+55+56+...+144. If you regroup s as (53+144)+(54+143)+...(98+99), then s=197+197+...+197. Part 4 197+197+...+197 is equal to the product of 197 and the number of pairs in the original sum. If there were 10 terms in the original sum, then there would be 5 pairs with a sum of 197. Therefore, s=197+197+...+197 would equal 197•5. Note that this method only works for sums with an even number of terms. Part 5 Determine how many terms there are in the original sum 53+54+55+56+...+144. Since each term is exactly one more than the last term, the difference between the first and last terms gives the number of terms after the first. For example, for the sum 53+54+55, the difference between the first and last terms is 2 and there are 2 terms after the first term. Part 6 Therefore, the number of terms in 53+54+55 is 2 additional terms+1 first term=3 terms for the whole sum. Of course, we didn't need to subtract to determine the number of terms in the simple sum 53+54+55, but finding a pattern in the simple example allows us to determine the number of terms in the much longer sum, 53+54+55+56+...+144. Part 7 There are (144−53)+1=92 terms in the given sum. Therefore, there are 922=46 pairs with a sum of 197. Multiply to find the sum s. s=46•197=9,062 Part 8 The sum of the sequence 53+54+55+56+...+144 is 9,062.
Question content area top Part 1 Can 0 be the identity for multiplication? Explain why or why not.
No, because if it were, then any number times 0 must be that number. The result in every case is 0, not the number multiplied by 0