1 gallon = 16 cups

2023 Handbook

2023 Test B Math · Question 42 of 57

Question

1 gallon = 16 cups

1 cup = 8 fluid ounces

If 1 gallon = $2^{x}$ fluid ounces, what is the value of x?

Show the answer

Correct answer: B

If 1 gallon is equal to 16 cups, and 1 cup is equal to 8 fluid ounces, then

$1\text{ gallon}=$

$(16)(8\text{ fluid ounces})=$

$128\text{ fluid ounces}$

$128=2^{7}$

$1\text{ gallon}=2^{7}\text{ fluid ounces}$

So, $x=7$ .

Question analysis

This question gives two facts, 16 cups in a gallon and 8 fluid ounces in a cup, and says a gallon equals 2 to the power x fluid ounces. Students find the exponent x. It combines a unit conversion with writing a number as a power of 2, and the exponent is easy to confuse with a multiplier.

NYS standards: 6.RP.3d converting measurement units
6.EE.1 whole-number exponent expressions
Grade level: Grade 6 content, by NYS standard
Compared with grade 7: At grade 7 Converting units with given facts is grade 6 work, and this conversion is a single multiplication. The twist is writing the result as a power of 2, which calls for fluency with powers that grade 7 work rarely asks for.
Estimated difficulty: Straightforward: two steps or one careful reading.
Thinking it demands: Depth of knowledge 2, skills and concepts · mainly apply
What makes it hard: Combines ideas minor
Something you must already know minor
Solution: 3 steps
Reading level: Question and choices: about grade 2, 16 words.
Students, last 30 days: Not enough answers yet to show (it takes 10).

Why it is hard, traps and wrong answers (shows the answer)

Why it is hard

The conversion is one multiplication, so the question really tests whether students can name the exponent for 128. Stopping at the cups, where 16 = 2⁴, or treating 2 to the power x as 2 times x gives a listed choice. Writing 16 as 2⁴ and 8 as 2³ and adding the exponents is the quickest route, and doubling from 1 seven times is a sure check.

Combines ideas minor Working it out
A unit conversion, 16 cups × 8 fluid ounces = 128 fluid ounces, has to be joined to exponents by recognizing 128 as a power of 2. (Embretson, 1998; Daniel & Embretson, 2010)
Something you must already know minor Working it out
Students must know or build the powers of 2 up to 128, and see that the exponent counts factors of 2, not the number of terms in a list starting at 1. (Anderson & Krathwohl, 2001)

Strategy: Convert 1 gallon to fluid ounces by multiplying the two facts, then keep halving the result, or keep doubling from 2, counting how many factors of 2 it takes.

Traps

Stops one step early: A choice is a number from partway through the solution.
A faulty procedure: A systematic but wrong procedure: adding fractions straight across, applying an exponent to each term of a sum, dropping a sign when moving a term.
Counting the ends: Inclusive or exclusive ends, fence posts, consecutive numbers: an answer that is one too many or too few, or an endpoint drawn open where it belongs closed (or the reverse).

The wrong answers, classified

A Stops one step early (King et al., 2004; Gierl et al., 2017)
Reveals partial knowledge Expected pull: moderate
Writes the 16 cups in a gallon as 2⁴ and stops before multiplying by the 8 fluid ounces in each cup, which adds 3 to the exponent.
C Counting the ends (Brown & Burton, 1978)
Reveals a slip Expected pull: weak
Lists the powers of 2 starting at 1, so counts 1, 2, 4, …, 128 as eight numbers and takes 8 as the exponent, though 1 is 2⁰.
D A faulty procedure (Brown & Burton, 1978; Tatsuoka, 1983)
Reveals a misconception Expected pull: moderate
Treats 2 to the power x as 2 times x, so 2x = 128 gives 64, confusing an exponent with a multiplier.

How the questions are analyzed, and the research behind it

Worked solution, step by step

Formulas used

Meaning of an exponent: $2^x = 2 \times 2 \times \cdots \times 2$; The 2 is the base, the number being multiplied. x is the exponent, the small raised number. It tells how many 2s are multiplied together. For example, 2 to the third power means 2 × 2 × 2, which is 8.
Multiplying powers of the same base: $2^a \times 2^b = 2^{a + b}$; a and b are exponents. The first power has a twos multiplied together. The second has b more twos. Multiplying puts all the twos together, so you add the exponents.

Sort out the question
A gallon is 16 cups. A cup is 8 fluid ounces. A gallon is also $2^x$ fluid ounces. You need x, the exponent. It tells how many 2s are multiplied together.
Change a gallon to fluid ounces
Each of the 16 cups holds 8 fluid ounces. So multiply.
$16 \times 8 = 128$
One gallon is 128 fluid ounces.
Write 128 as a power of 2
Now you need $2^x = 128$ . Keep multiplying by 2, and count the 2s.
Number of 2s 1 2 3 4 5 6 7
Product 2 4 8 16 32 64 128
It takes seven 2s to make 128.
$2^7 = 128$
So x = 7.
Give the answer
Watch out: A gallon is 16 cups, and $16 = 2^4$ . Stopping there gives 4, choice A. And $2^x$ does not mean 2 times x. Solving 2 × x = 128 gives 64, choice D.
So the answer is B, 7.

Check your answer

Use the exponent rule. 16 is $2^4$ and 8 is $2^3$ . So $16 \times 8 = 2^4 \times 2^3 = 2^{4 + 3} = 2^7$ . The exponent is 7 again.

The original DOE question

Question 99 as printed on page 203 of the DOE handbook. — As printed in the DOE’s NYC Guide to the Specialized High Schools Admissions Test (SHSAT) for 2024 Admissions, page 203, question 99. Open page 203 of the handbook (PDF)

Number of 2s	1	2	3	4	5	6	7
Product	2	4	8	16	32	64	128