Download MATH 302A Sample Test Questions with Solutions: 1. If the pattern

MATH 302A Sample Test Questions with Solutions:
1. If the pattern continues, what is the 50th digit in the following: 101001000100001...
To continue the pattern, put one additional 0 before the digit 1. So, the digit “1”
shows up in the 1st, 3rd, 6th, 10th, 15th, … place. To continue to find where the 1s are,
continue this pattern: the 21st, 28th, 36th, 45th, and 55th place are where the 1s are. Since the
50th digit is not a 1, it must be a 0.
2. What is the digit in the ones position if you raise 6 to the 50th power?
6 = 61 = 6
6 ⋅ 6 = 62 = 36
6 ⋅ 6 ⋅ 6 = 63 = 216
Each time you multiply 6 by a number with a 6 in the ones place, you get a new
number with a 6 in the ones place. So, 6 raised to the 50th power is a number with a 6 in the
ones place.
3. Hannah is playing a game in which it is possible to score 0 points, 3 points, 4 points, or 8
points each round. After 3 rounds of play, what are all possible total scores that
Hannah might have?
Possible scores are:
0 (0 + 0 + 0), 3 (0 + 0 + 3), 4 (0 + 0 + 4), 6 (0 + 3 + 3), 7 (0 + 3 + 4), 8 (0 + 4 + 4),
9 (3 + 3 + 3), 10 (3 + 3 + 4), 11 (3 + 4 + 4), 12 (4 + 4 + 4), 14 (3 + 3 + 8),
15 (3 + 4 + 8), 16 (4 + 4 + 8), 19 (3 + 8 + 8), 20 (4 + 8 + 8), 24 (8 + 8 + 8)
Note that there is more than one way to get some of these scores.
4. Consider this sequence: 3, 8, 18, 38, 78, 158, . . .
a) What is the next term?
b) What is the 20th term?
a) Each time, you are adding multiples of 5: first 5 · 1, then 5 • 2, then 5 • 4,
then 5 • 8, ... So, the 7th term is 158 + 5 • 32, which is 318.
b) The 20th term is 2,621,438.
The general formula is 3 + 5 • (2n-1 – 1) where n is the number of the term.
Therefore, the 20th term is 3 + 5 • (2n-1 – 1) = 3 + 5 • (219 – 1) = 2621438
3 + 5 • (524288 – 1) = 3 + 5 • 524287 = 2,621,438.
This is a difficult general formula to determine, so on the test, I would give you
part a but not part b or I would give you a sequence with a simpler general
formula.
5. Use a diagram to illustrate that the sum of an odd number and an odd number is an even
number.
▲▲▲▲▲
▲▲▲▲▲▲▲
▲▲▲▲▲▲
▲▲▲▲▲▲▲▲
Every even number can be made by arranging two equal rows of objects. Every odd
number can be made by arranging two rows, with the first row having exactly one more
object than the second row.
When adding two odd numbers, the two left over ones will make a pair, and so two
equal rows can be formed.
6. Show a number line model and a picture model for the addition problem 8 + 5 = 13
7. Arrange in order, smallest first.
Mayan:
7 • 20 + 10 = 150;
Babylonian: 2 • 60 + 1 = 121;
Egyptian:
3 • 100 + 4 • 10 + 1 = 341.
So, in order from smallest to greatest: Babylonian, Mayan, and Egyptian
8. How is the 7 in 307 different from the 7 in 4172?
In 307, the 7 represents 7 units, or 7 ones, or just 7. In 4172, the 7 represents 7
tens, or 70. The place value of the digit determines the number it represents.
9. a) What is the largest 4-digit number that is possible in base 6?
b) What is the base 10 value of the number?
a) The largest 4-digit number that is possible in base 6 is 55556.
b) The base 10 value of 55556 is [5 × (6 × 6 × 6)] + [5 × (6 × 6)] + [5 × 6] + 5 = 1295
10. Which is smaller, 2547 or 12235?
2547 = [2 × (7 × 7)] + [5 × 7] + 4 = 137
and
12235 = [1 × (5 × 5 × 5)] + [2 × (5 × 5)] + [2 × 5] + 3 = 188
Therefore, 12235 is greater than 2545.
11. Use mental arithmetic to add 749 +801 +435. Describe your method for finding the sum.
12. Explain the mistake made by a child in this problem. What aspect of subtraction or place
value is the child not applying or applying incorrectly?
76
− 48
22
The child subtracted 6 from 8 instead of using the 10 he or she took from the tens
place and combining it with the 6 units so the 8 units could be subtracted.
11. The number line that illustrates 4 – 5 = -1 is D.
MATH 302A Sample Test Questions with Solutions:
1. If the pattern continues, what is the 50th digit in the following: 101001000100001...
To continue the pattern, put one additional 0 before the digit 1. So, the digit “1”
shows up in the 1st, 3rd, 6th, 10th, 15th, … place. To continue to find where the 1s are,
continue this pattern: the 21st, 28th, 36th, 45th, and 55th place are where the 1s are. Since the
50th digit is not a 1, it must be a 0.
2. What is the digit in the ones position if you raise 6 to the 50th power?
6 = 61 = 6
6 ⋅ 6 = 62 = 36
6 ⋅ 6 ⋅ 6 = 63 = 216
Each time you multiply 6 by a number with a 6 in the ones place, you get a new
number with a 6 in the ones place. So, 6 raised to the 50th power is a number with a 6 in the
ones place.
3. Hannah is playing a game in which it is possible to score 0 points, 3 points, 4 points, or 8
points each round. After 3 rounds of play, what are all possible total scores that
Hannah might have?
Possible scores are:
0 (0 + 0 + 0), 3 (0 + 0 + 3), 4 (0 + 0 + 4), 6 (0 + 3 + 3), 7 (0 + 3 + 4), 8 (0 + 4 + 4),
9 (3 + 3 + 3), 10 (3 + 3 + 4), 11 (3 + 4 + 4), 12 (4 + 4 + 4), 14 (3 + 3 + 8),
15 (3 + 4 + 8), 16 (4 + 4 + 8), 19 (3 + 8 + 8), 20 (4 + 8 + 8), 24 (8 + 8 + 8)
Note that there is more than one way to get some of these scores.
4. Consider this sequence: 3, 8, 18, 38, 78, 158, . . .
a) What is the next term?
b) What is the 20th term?
a) Each time, you are adding multiples of 5: first 5 · 1, then 5 • 2, then 5 • 4,
then 5 • 8, ... So, the 7th term is 158 + 5 • 32, which is 318.
b) The 20th term is 2,621,438.
The general formula is 3 + 5 • (2n-1 – 1) where n is the number of the term.
Therefore, the 20th term is 3 + 5 • (2n-1 – 1) = 3 + 5 • (219 – 1) = 2621438
3 + 5 • (524288 – 1) = 3 + 5 • 524287 = 2,621,438.
This is a difficult general formula to determine, so on the test, I would give you
part a but not part b or I would give you a sequence with a simpler general
formula.
5. Use a diagram to illustrate that the sum of an odd number and an odd number is an even
number.
▲▲▲▲▲
▲▲▲▲▲▲▲
▲▲▲▲▲▲
▲▲▲▲▲▲▲▲
Every even number can be made by arranging two equal rows of objects. Every odd
number can be made by arranging two rows, with the first row having exactly one more
object than the second row.
When adding two odd numbers, the two left over ones will make a pair, and so two
equal rows can be formed.
6. Show a number line model and a picture model for the addition problem 8 + 5 = 13
7. Arrange in order, smallest first.
Mayan:
7 • 20 + 10 = 150;
Babylonian: 2 • 60 + 1 = 121;
Egyptian:
3 • 100 + 4 • 10 + 1 = 341.
So, in order from smallest to greatest: Babylonian, Mayan, and Egyptian
8. How is the 7 in 307 different from the 7 in 4172?
In 307, the 7 represents 7 units, or 7 ones, or just 7. In 4172, the 7 represents 7
tens, or 70. The place value of the digit determines the number it represents.
9. a) What is the largest 4-digit number that is possible in base 6?
b) What is the base 10 value of the number?
a) The largest 4-digit number that is possible in base 6 is 55556.
b) The base 10 value of 55556 is [5 × (6 × 6 × 6)] + [5 × (6 × 6)] + [5 × 6] + 5 = 1295
10. Which is smaller, 2547 or 12235?
2547 = [2 × (7 × 7)] + [5 × 7] + 4 = 137
and
12235 = [1 × (5 × 5 × 5)] + [2 × (5 × 5)] + [2 × 5] + 3 = 188
Therefore, 12235 is greater than 2545.
11. Use mental arithmetic to add 749 +801 +435. Describe your method for finding the sum.
12. Explain the mistake made by a child in this problem. What aspect of subtraction or place
value is the child not applying or applying incorrectly?
76
− 48
22
The child subtracted 6 from 8 instead of using the 10 he or she took from the tens
place and combining it with the 6 units so the 8 units could be subtracted.
11. The number line that illustrates 4 – 5 = -1 is D.
MATH 302A Sample Test Questions with Solutions:
1. If the pattern continues, what is the 50th digit in the following: 101001000100001...
To continue the pattern, put one additional 0 before the digit 1. So, the digit “1”
shows up in the 1st, 3rd, 6th, 10th, 15th, … place. To continue to find where the 1s are,
continue this pattern: the 21st, 28th, 36th, 45th, and 55th place are where the 1s are. Since the
50th digit is not a 1, it must be a 0.
2. What is the digit in the ones position if you raise 6 to the 50th power?
6 = 61 = 6
6 ⋅ 6 = 62 = 36
6 ⋅ 6 ⋅ 6 = 63 = 216
Each time you multiply 6 by a number with a 6 in the ones place, you get a new
number with a 6 in the ones place. So, 6 raised to the 50th power is a number with a 6 in the
ones place.
3. Hannah is playing a game in which it is possible to score 0 points, 3 points, 4 points, or 8
points each round. After 3 rounds of play, what are all possible total scores that
Hannah might have?
Possible scores are:
0 (0 + 0 + 0), 3 (0 + 0 + 3), 4 (0 + 0 + 4), 6 (0 + 3 + 3), 7 (0 + 3 + 4), 8 (0 + 4 + 4),
9 (3 + 3 + 3), 10 (3 + 3 + 4), 11 (3 + 4 + 4), 12 (4 + 4 + 4), 14 (3 + 3 + 8),
15 (3 + 4 + 8), 16 (4 + 4 + 8), 19 (3 + 8 + 8), 20 (4 + 8 + 8), 24 (8 + 8 + 8)
Note that there is more than one way to get some of these scores.
4. Consider this sequence: 3, 8, 18, 38, 78, 158, . . .
a) What is the next term?
b) What is the 20th term?
a) Each time, you are adding multiples of 5: first 5 · 1, then 5 • 2, then 5 • 4,
then 5 • 8, ... So, the 7th term is 158 + 5 • 32, which is 318.
b) The 20th term is 2,621,438.
The general formula is 3 + 5 • (2n-1 – 1) where n is the number of the term.
Therefore, the 20th term is 3 + 5 • (2n-1 – 1) = 3 + 5 • (219 – 1) = 2621438
3 + 5 • (524288 – 1) = 3 + 5 • 524287 = 2,621,438.
This is a difficult general formula to determine, so on the test, I would give you
part a but not part b or I would give you a sequence with a simpler general
formula.
5. Use a diagram to illustrate that the sum of an odd number and an odd number is an even
number.
▲▲▲▲▲
▲▲▲▲▲▲▲
▲▲▲▲▲▲
▲▲▲▲▲▲▲▲
Every even number can be made by arranging two equal rows of objects. Every odd
number can be made by arranging two rows, with the first row having exactly one more
object than the second row.
When adding two odd numbers, the two left over ones will make a pair, and so two
equal rows can be formed.
6. Show a number line model and a picture model for the addition problem 8 + 5 = 13
7. Arrange in order, smallest first.
Mayan:
7 • 20 + 10 = 150;
Babylonian: 2 • 60 + 1 = 121;
Egyptian:
3 • 100 + 4 • 10 + 1 = 341.
So, in order from smallest to greatest: Babylonian, Mayan, and Egyptian
8. How is the 7 in 307 different from the 7 in 4172?
In 307, the 7 represents 7 units, or 7 ones, or just 7. In 4172, the 7 represents 7
tens, or 70. The place value of the digit determines the number it represents.
9. a) What is the largest 4-digit number that is possible in base 6?
b) What is the base 10 value of the number?
a) The largest 4-digit number that is possible in base 6 is 55556.
b) The base 10 value of 55556 is [5 × (6 × 6 × 6)] + [5 × (6 × 6)] + [5 × 6] + 5 = 1295
10. Which is smaller, 2547 or 12235?
2547 = [2 × (7 × 7)] + [5 × 7] + 4 = 137
and
12235 = [1 × (5 × 5 × 5)] + [2 × (5 × 5)] + [2 × 5] + 3 = 188
Therefore, 12235 is greater than 2545.
11. Use mental arithmetic to add 749 +801 +435. Describe your method for finding the sum.
12. Explain the mistake made by a child in this problem. What aspect of subtraction or place
value is the child not applying or applying incorrectly?
76
− 48
22
The child subtracted 6 from 8 instead of using the 10 he or she took from the tens
place and combining it with the 6 units so the 8 units could be subtracted.
11. The number line that illustrates 4 – 5 = -1 is D.
MATH 302A Sample Test Questions with Solutions:
1. If the pattern continues, what is the 50th digit in the following: 101001000100001...
To continue the pattern, put one additional 0 before the digit 1. So, the digit “1”
shows up in the 1st, 3rd, 6th, 10th, 15th, … place. To continue to find where the 1s are,
continue this pattern: the 21st, 28th, 36th, 45th, and 55th place are where the 1s are. Since the
50th digit is not a 1, it must be a 0.
2. What is the digit in the ones position if you raise 6 to the 50th power?
6 = 61 = 6
6 ⋅ 6 = 62 = 36
6 ⋅ 6 ⋅ 6 = 63 = 216
Each time you multiply 6 by a number with a 6 in the ones place, you get a new
number with a 6 in the ones place. So, 6 raised to the 50th power is a number with a 6 in the
ones place.
3. Hannah is playing a game in which it is possible to score 0 points, 3 points, 4 points, or 8
points each round. After 3 rounds of play, what are all possible total scores that
Hannah might have?
Possible scores are:
0 (0 + 0 + 0), 3 (0 + 0 + 3), 4 (0 + 0 + 4), 6 (0 + 3 + 3), 7 (0 + 3 + 4), 8 (0 + 4 + 4),
9 (3 + 3 + 3), 10 (3 + 3 + 4), 11 (3 + 4 + 4), 12 (4 + 4 + 4), 14 (3 + 3 + 8),
15 (3 + 4 + 8), 16 (4 + 4 + 8), 19 (3 + 8 + 8), 20 (4 + 8 + 8), 24 (8 + 8 + 8)
Note that there is more than one way to get some of these scores.
4. Consider this sequence: 3, 8, 18, 38, 78, 158, . . .
a) What is the next term?
b) What is the 20th term?
a) Each time, you are adding multiples of 5: first 5 · 1, then 5 • 2, then 5 • 4,
then 5 • 8, ... So, the 7th term is 158 + 5 • 32, which is 318.
b) The 20th term is 2,621,438.
The general formula is 3 + 5 • (2n-1 – 1) where n is the number of the term.
Therefore, the 20th term is 3 + 5 • (2n-1 – 1) = 3 + 5 • (219 – 1) = 2621438
3 + 5 • (524288 – 1) = 3 + 5 • 524287 = 2,621,438.
This is a difficult general formula to determine, so on the test, I would give you
part a but not part b or I would give you a sequence with a simpler general
formula.
5. Use a diagram to illustrate that the sum of an odd number and an odd number is an even
number.
▲▲▲▲▲
▲▲▲▲▲▲▲
▲▲▲▲▲▲
▲▲▲▲▲▲▲▲
Every even number can be made by arranging two equal rows of objects. Every odd
number can be made by arranging two rows, with the first row having exactly one more
object than the second row.
When adding two odd numbers, the two left over ones will make a pair, and so two
equal rows can be formed.
6. Show a number line model and a picture model for the addition problem 8 + 5 = 13
7. Arrange in order, smallest first.
Mayan:
7 • 20 + 10 = 150;
Babylonian: 2 • 60 + 1 = 121;
Egyptian:
3 • 100 + 4 • 10 + 1 = 341.
So, in order from smallest to greatest: Babylonian, Mayan, and Egyptian
8. How is the 7 in 307 different from the 7 in 4172?
In 307, the 7 represents 7 units, or 7 ones, or just 7. In 4172, the 7 represents 7
tens, or 70. The place value of the digit determines the number it represents.
9. a) What is the largest 4-digit number that is possible in base 6?
b) What is the base 10 value of the number?
a) The largest 4-digit number that is possible in base 6 is 55556.
b) The base 10 value of 55556 is [5 × (6 × 6 × 6)] + [5 × (6 × 6)] + [5 × 6] + 5 = 1295
10. Which is smaller, 2547 or 12235?
2547 = [2 × (7 × 7)] + [5 × 7] + 4 = 137
and
12235 = [1 × (5 × 5 × 5)] + [2 × (5 × 5)] + [2 × 5] + 3 = 188
Therefore, 12235 is greater than 2545.
11. Use mental arithmetic to add 749 +801 +435. Describe your method for finding the sum.
12. Explain the mistake made by a child in this problem. What aspect of subtraction or place
value is the child not applying or applying incorrectly?
76
− 48
22
The child subtracted 6 from 8 instead of using the 10 he or she took from the tens
place and combining it with the 6 units so the 8 units could be subtracted.
11. The number line that illustrates 4 – 5 = -1 is D.