Sample Test Questions:
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.
What is the digit in the one’s position if you raise 6 to the 50th power?
61 = 6
62 = 36
63 = 216
Each time you multiply 6 by a number with a 6 in the one’s place, you get a new number with a 6 in the one’s place. So, 6 raised to the 50th power is a number with a 6 in the one’s place.
Gwen had 18 coins in her pocket that had a total value of $1.25. Find two different combinations of coins that could have been in Gwen’s pocket.
One possible answer is:Gwen has 1 quarter, 7 dimes, 5 nickels, and 5 pennies (25 + 70 + 25 + 5 = 125) in her pocket. Another possibility is that Gwen has 7 dimes and 11 nickels (70 + 55 = 125).
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=00+0+3=30+0+4=40+3+3=6
0 + 3 + 4=70 + 4 + 4=83+3+3=93+3+4=10 3+4+4=11 4+4+4=12 3 + 3 + 8=14 3 + 4 + 8=15
4 + 4 + 8=163 + 8 + 8=194 + 8 + 8=208 + 8 + 8=24
Note: Some of these scores could be obtained in other ways.
Consider this sequence: 3, 8, 18, 38, 78, 158, . . .
a) What is the next term?
b) What is the 20th term?
Each time, you are adding multiples of 5, first 5 ∙ 1, then 2 • 5, then 4 • 5, then 8 • 5, ... So, the 7th term is 158 + 32 • 5, which is 318. The 20thterm 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 • (219 – 1) = 3 + 5 • (524288 – 1) =
3 + 5 • (524288 – 1) = 3 + 5 • 524,287 = 2,621,438.
Note: This is a pretty difficult general formula to find. On the test, I might give ask you for part a) but not part b).
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 moreobject than the second row. When you put the two rows together, the first row still hasexactly one more object than the second row, and so it is an odd number.
Example:
Even (always makes 2 equal rows) Odd (always makes 2 equal rows
with one left over.) When adding two odd numbers, the two left over ones will make a pair, and so two equal rows canbe formed.
Don bought some rose bushes that cost $9.00 each, and Joyce bought some lilac bushes that cost $14.00 each. If together they spent $82 on these bushes and they bought more rose bushes thanlilac bushes, how many of each did they buy?
We know Don and Joyce bought $82 worth of flowers. They purchased rose bushes for $9 each and lilac bushes for $14 each. We don’t know how many of eachthey bought, so let’s try to get exactly $82.
If they bought 5 roses, that is 5 • 9 = 45, and 82 – 45 = 37. So, they can’t get exactly 3 lilacs. If they bought 6 roses, that is 6 • 9 = 54, and 82 – 54 = 28, they can get exactly 2 lilacs. So, Don and Joyce bought six rose bushes and two lilac bushes.
A carpenter is cutting wood strips to make picture frames. Each frame requires 2 pieces that are 8 inches long and two pieces that are 10 inches long. How many frames can the carpenter cut froma piece of wood that is 8 feet long? How much waste will there be?
Each frame requires two sides that are 10 inches and two sides that are 8 inches. That is a totalof 36 inches. Thirty-six inches is the same as 3 feet, and so this carpenter can make two completeframes, and he or she will have 2 feet left over.
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
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 7represents 7 tens, or 70. The digit alone is 7. The place value for the firstnumber shows 7 in the ones place. The place value for the second numbershows 7 in the tens place.
The number line that illustrates 4 – 5 = -1 is D.