5.Wally kept track of last week's money transactions. His salary was $150 plus $54 in overtime and
$260 in tips. His transportation expenses were $22, his food expenses were $60, his laundry costs
were $15, his entertainment expenditures were $58, and his rent was $185.
Total Income = $150 + $54 + $260 = $464
Total expense = $22 + $60 + $15 +$58 + $185 = $340
Balance = $464 - $340 = $124
9. Analyze the following computations. Explain what is wrong in each case. a. 28+75=913
8 + 5 = 13 .
2+7=9
Then got 913
The 1 in 13 is to be added to 9 to get 10
The correct answer is 103
b. 28+75=121
7 + 5 = 12
8-7 =1
Then got 121
The correct answer is 103
c.305-259=154
9-5=4
5-0 =5
3-2=1
Then got 154
This procedure is wrong.
The correct answer is 305-259 = 46
d. 305-259=56
15 – 9 = 6
30-25 =5
Then got 56
The correct answer is 305 – 259 = 46
10. Give reasons for each of the following steps: 16+31=(1×10+6)+(3×10+1) =(1×10+3×10)+(6+1)
=(1+3)10+(6+1) =4×10+7 =47
Reason: 16 contains 1 ten and 6 units. 16 = 10 + 6 = 1
31 contains 3 tens and I unit. 31 =
Reason: the tens and the units are grouped using commutative property and associative property of
addition.
Distributive property of addition and multiplication is used.
1+3=4 and 6+1=7
4 tens and 7 units is 47.
6.Place parentheses, if needed, to make each of the following equations true:
5+6×3=33
Interchange and
8+7-3=12
6+8-2/2=13
9+6/3=5
22.Millie and Samantha began saving money at the same time. Millie plans to save $3 a month, and
Samantha plans to save $5 a month. After how many months will Samantha have saved exactly $10
more than Millie?
Month 1 2 3 4 5
Millie 3 3+3=6 6+3=9 9+3=12 12+3=15
Samantha 5 5+5=10 10+5=15 15+5=20 20+5=25
After 5 months Millie will have $15 savings and Samantha will have $25 savings. So after 5 months
Samantha have saved exactly $10 more than Millie.
17.Gina buys apples from an orchard and then sells them at a country fair in bags of 3 for $1 a bag.
She bought 50 boxes of apples, 36 apples in a box, and paid $452. If she sold all but 18 apples, what
was her total profit?
Cost price =$452
Number apples purchased =
Number of apples sold = 1800 – 18 = 1782
Cost price of 1782 apples = $452
Number bags sold =
Selling Price of 1782 apples = $1
Profit = $594-$447.48=$146.52
19a.Give reasons for each of the following steps: 56×10=(5×10+6)×10 =(5×10)×10+6×10
=5×(10×10)+6×10 = 5×102 + 6×10 =5×102=6×10+0×1 =560
(56 is 5 tens and 6 units)
Use distributive property
Use associative property
is
is 100
5 hundreds and 6 tens is 560
7.Compute each of the following mentally. In each case, briefly explain your method.
86+37
97+54
230+60+70+44+40+6
10.Use front-end estimation with adjustment to estimate each of the following:
2215+3023+5987+975
234+478+987+319+469
1.A number leaves remainder 6 when divided by 10. What is the remainder when the number is
divided by 5? Justify your reasoning.
The remainder is 1
Reason: number = multiple of 10 + 6 = multiple of 5 + 5 +1=multiple of 5 +1
1. What is the difference between mental mathematics and computational estimation?
In mental mathematics is the computation is done mentally using similarities discovered and stored
in the mind. Computational estimation is computing using systematically the rules of mathematics.
5. Give several examples from real-world situations where an estimate, rather than an exact answer,
is sufficient.
A student getting more than 50% marks is considered as an above average student. So to determine
whether a student is above average, an estimate of the marks is enough.
16. For which integers a, b, and c does a-b-c=a-(b-c)? Justify your answer.
Suppose
Then
Then –
Then
If c=0, the condition is satisfied.
19. Find all integers x, if there are any, such that the following are true: a.-x is positive.
b.-x is negative.
c.-x-1 is positive.
is positive if x is greater than 1. That is x=2,3,…
d. |x|=2 15.
if and only if or
A fourth-grade student devised the following subtraction algorithm for subtracting 84-27: 4 minus 7
equals negative 3. 84-27=-3 80 minus 20 equals 60. 84-27=-3 60 60 plus negative 3 equals 57. 84-
27=-3+60=57 Thus the answer is 57. What is your response as a teacher?
His procedure is correct.
12. Show that the distributive property of multiplication over addition, a(b+c)=ab+ac, is true for
each of the following values of a, b, and c:
a=-1,b=-5,c=-2
a=-3, b=-3, c=2
13. Compute each of the following:
a.(-2)3
b. (-2)4
c.(-10)5÷(-10)2
d.(-3)5÷(-3)
e.(-1)50
f. (-1)151
g.-2+3*5-1
h. 10-3*7-4(-2)+3 17.
Identify the property of integers being illustrated in each of the following:
a.(-3)(4+5)=(4+5)(-3)
commutative property
b.(-4)(-7)?I
c.5[4(-3)]=(5*4)(-3)
Associative property
d.(-9)[5+(-8)]=(-9)*5+(-9))-8)
Disributive Property
5. Hosni gave the following argument that -(a+b)=-a+-b for all integers a and b. If the argument is
correct, supply the missing reasons. If it is incorrect, explain why. -(a+b)=(-1)(a+b) =(-1)a+(-1)b =-a+-b
3. Without using a calculator, test each of the following numbers for divisibility by 2, 3, 4, 5, 6, 8, 9,
10, and 11:
a. 746,988
Divisible by 2 : digit in unit’s place is 8
Sum of the digits = 7+4+6+9+8+8=6 is a multiple of 3
Divisible by 3
Divisible by 2 and divisible by 3 implies divisible by 6
Last 2 digits 88 is divisible by 4, so the number is divisible by 4
Last three digits 988 is not divisible by 8 , so not divisible by 8
Sum of the digits is not a multiple of 9 and so not divisible by 9
The last digit neither 0 nor 5, so not divisible by 5 or 10
Collect digits 3 at a time
746
988
Difference is 242
242 is a multiple of 11, so the number is divisible by 11
242 is not a multiple of 7 and so the number is not divisible by 7
b. 81,342
Divisible by 2 : digit in unit’s place is 2
Sum of the digits = 8+1+3+4+2=9is a multiple of 3
Divisible by 3
Divisible by 2 and divisible by 3 implies divisible by 6
Last 2 digits 42 is not divisible by 4, so the number is not divisible by 4
So not divisible by 8
Sum of the digits is a multiple of 9 and so divisible by 9
The last digit neither 0 nor 5, so not divisible by 5 or 10
Collect digits 3 at a time
81
342
Difference is 261
261 is not a multiple of 11, so the number is not divisible by 11
261 is not a multiple of 7 and so the number is not divisible by 7
c. 15,810
Divisible by 2, 5 and 10
Not divisible by 4 or 8
Divisble by 3 but not by 9
Divisible by 6
Not divisible by 7 or 11
17. Using only divisibility tests, explain whether 6,868,395 is divisible by 15.
Last digit is 5
Divisible by 5
Sum of the digits is 9
Divisble by 3
Divisible by 15
3. Is the area of each of the following rectangles divisible by 4? Explain why or why not. 52,832 cm
324,518 cm 52,834 cm 324,514 cm
The area is divisible by 4 because the sides are divisible by 2
4. Can you find three consecutive natural numbers none of which is divisible by 3? Explain your
answer.
It is not possible. Every 3rd natural number is divisible by 3.
6. Find the prime factorizations of the following: a. 1*2*3*4*5*6*7*8*9*10
b. 102*26*4910
c. 251
d. 1001
9. Mr. Arboreta wants to plant fruit trees in a rectangular array. For each of the following numbers
of trees, find all possible numbers of rows if each row is to have the same number of trees:
a. 36
b. 28
c. 17
d. 144
1. Explain why the product of any three consecutive integers is divisible by 6.
Every second number is divisible by 2 and every 3rd number is divisible by 3. The product
three numbers , one of which is divisible by 2 and another divisible by 3 is divisible by 6.
11. Three motorcyclists ride around a circular race course starting at the same place and the same
time. The first passes the starting point every 12 min, the second every 18 min, and the third every
16 min. After how many minutes will all three pass the starting point again at the same time? Explain
your reasoning.
The first cyclist passes the starting point at the times
12, 24, 36, 48, 60, 72,84,96,108,120,132,144, 156 ….
The second cyclist passes the starting point at the times
18, 36, 54,, 72,90,108, 126, 144, 162 ….
The third cyclist passes the starting point at the times
16, 32, 48, 64, 80, 96, 112, 128, 144, 160….
The least common multiple of 12, 18 and 16 is 144
After 144 minutes all three of them pass the starting point.
18. At the Party Store, paper plates come in packages of 30, paper cups in packages of 15, and
napkins in packages of 20. What is the least number of plates, cups, and napkins that can be
purchased so that there is an equal number of each?
LCM of 30, 15 and 20 is calculated as
LCM=
60 is the required number.