Student edition of Test 26
It preserves the twelve-question order, base notation, construction conditions, and original point values.
Lesson 26.7 · Student exercise
Chapter 26 Exercise
Test 26: Decimal and Binary. Read every base label carefully, expand each numeral by its own place values, and use take-or-leave binary choices when a construction problem asks for exact quantities.
Exercise directions
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Scoring notes
How to read this exercise
Base labels matter. A subscript 10 means decimal; a subscript 2 means binary. The same written digits can have different values in different bases.
Spaces are only separators. In a form such as 10 0012, the space is included only to make the digits easier to read.
Questions 4 and 5 ask for expansions. For this learning edition, show each digit multiplied by its place value, including zero terms. Omitting zero terms gives an equivalent expression, but showing them helps explain each place. You may type powers using ^, for example 6*10^2 + 2*10^1 + 0*10^0. Do not replace the requested expansion with only a converted final value.
Use only legal base-2 digits. Unnecessary leading zeros may be removed, but internal zeros must remain in their correct places.
Whole boxes stay closed. In Questions 10 and 11, each box contains a positive whole-number quantity, and a request is filled by taking complete boxes only.
Question 12 uses subsets. Select at least one of the six given numbers; each number may be used at most once, and changing the order of the same selection does not make a new sum.
Section I — Fill In
Questions 1–9 · 5 points each · 45 points
Identify the legal binary digits
5 pointsIn the binary system, which digits can appear in a numeral? List every permitted digit once, separated by commas.
State the binary carrying rule
5 pointsIn binary, one is carried to the next place whenever the current place reaches how many units?
Evaluate powers of two
5 pointsComplete all three values.
Expand decimal numerals by powers of ten
5 pointsRead the request carefully: write a base-10 place-value expansion. This question is not asking for binary conversion.
Expand binary numerals by powers of two
5 pointsShow the expansion: write each bit multiplied by its power of two. Do not write only the decimal value.
Convert binary numerals to decimal
5 pointsRewrite each binary numeral as a decimal numeral.
Convert decimal numerals to binary
5 pointsRewrite each decimal numeral as a binary numeral.
Add in binary
5 pointsCalculate and give the result in binary:
(10 110)2 + (1010)2
Subtract in binary
5 pointsCalculate and give the result in binary:
(1 101 010)2 − (10 101)2
Section II — Extended Response
Questions 10–12 · 25 points each · 75 points
Pack 1,000 rounds into exactly ten boxes
25 pointsA warehouse custodian packs 1,000 rounds into exactly 10 sealed boxes. For any requested whole number from 1 through 1,000, the custodian must be able to hand over one or more unopened boxes whose contents total exactly that request. How many rounds should be placed in each box?
These labels mark the ten box positions; enter their quantities in the answer area below.
Need a hint?
Start with a smaller problem. If your boxes can make every total from 0 to R, adding a box containing b items gives totals b through b+R too. To avoid a gap, b must be at most R+1. Include 0 (taking no boxes) when counting choices: k boxes have at most 2^k selections. Explain coverage of every request, not just the total packed.
Design exact-choice boxes for 500 eggs
25 points500 pigeon eggs are packed into boxes. A customer asking for any whole number from 1 through 500 must be served exactly by taking complete unopened boxes. For this learning edition, also find the fewest possible boxes. Describe how to pack the eggs and state how many boxes are needed. Different packings are acceptable if you prove that they satisfy every request and use the minimum number of boxes.
Need a hint?
Use the same no-gap argument as Question 10. Count the possible selections, including the empty selection for zero, to prove a lower bound on the number of boxes. Then build a packing that reaches that bound and totals exactly 500. Different packings may work; justify yours.
Find the 39th powers-of-three subset sum
25 pointsUse the six numbers:
13⁰
33¹
93²
273³
813⁴
2433⁵
Choose a nonempty subset and add its members, using each number at most once. The 63 resulting sums are placed in increasing order:
1, 3, 4, 9, 10, 12, 13, 27, …
What is the 39th number in this ordered list?
Need a hint?
Try the smaller list 1, 3, 9. Its nonempty sums in increasing order are 1, 3, 4, 9, 10, 12, 13; the selection masks (in the order 9, 3, 1) are 001, 010, 011, 100, 101, 110, 111. Each new power of 3 exceeds the sum of all smaller ones, so the highest changed selection decides the order. Explain how this connects a rank to a binary selection mask.