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\(\mathrm{a = 0.73 \times 10^{-5}}\), \(\mathrm{b = \frac{8}{100,000}}\), \(\mathrm{c = \frac{0.046}{100}}\), \(\mathrm{d = \frac{1}{2}(0.00001 + 0.0001)}\)
Which of the following must be true of the numbers shown above?
Let's start by understanding what we need to do. We have four different numbers given in different formats, and we need to figure out which one is smallest, which is next smallest, and so on until we get the largest one.
Think of it like having four boxes with different labels showing amounts, but the labels are written in different ways - some as fractions, some with scientific notation, some as decimals. To compare them fairly, we need to convert them all to the same format so we can easily see which amount is actually bigger or smaller.
Our four numbers are:
Process Skill: TRANSLATE - Converting different mathematical expressions into comparable forms
Now let's convert each expression into regular decimal numbers that we can easily compare.
For a = \(0.73 \times 10^{-5}\):
When we multiply by \(10^{-5}\), we're moving the decimal point 5 places to the left.
\(0.73 \times 10^{-5} = 0.00000073\)
For b = \(\frac{8}{100,000}\):
This is 8 divided by 100,000.
b = \(8 \div 100,000 = 0.00008\)
For c = \(\frac{0.046}{100}\):
This is 0.046 divided by 100.
c = \(0.046 \div 100 = 0.00046\)
For d = \(\frac{1}{2}(0.00001 + 0.0001)\):
First, let's add what's in the parentheses: \(0.00001 + 0.0001 = 0.00011\)
Then multiply by 1/2: d = \(\frac{1}{2} \times 0.00011 = 0.000055\)
Process Skill: SIMPLIFY - Breaking down complex expressions into manageable steps
Now we have all our numbers in decimal form:
To compare these easily, let's look at the first non-zero digit in each:
Since c has its first non-zero digit furthest to the left, it's the largest.
Between b and d (both have first non-zero digits in 5th place): 0.00008 vs 0.000055, so b > d.
Finally, a is smallest since its first non-zero digit is in the 7th decimal place.
Therefore, from smallest to largest: \(a < d < b < c\)
Looking at our result \(a < d < b < c\), we need to find this exact ordering in the answer choices.
Checking each option:
The correct answer is E. \(a < d < b < c\)
Our decimal conversions showed:
a = 0.00000073 < d = 0.000055 < b = 0.00008 < c = 0.00046
1. Misunderstanding scientific notation conversion
Students often struggle with \(10^{-5}\) and may incorrectly think it means multiplying by -5 or moving the decimal point in the wrong direction. They might convert \(0.73 \times 10^{-5}\) to \(0.73 \times (-5) = -3.65\) instead of recognizing that \(10^{-5}\) means \(\frac{1}{100,000}\) and requires moving the decimal point 5 places to the left.
2. Confusion about order of operations in compound expressions
For expression d = \(\frac{1}{2}(0.00001 + 0.0001)\), students may incorrectly distribute the 1/2 to each term first, calculating \(\frac{1}{2}(0.00001) + \frac{1}{2}(0.0001)\), or they may forget to perform the addition inside parentheses first before multiplying by 1/2.
3. Inconsistent conversion strategy
Students may attempt to compare numbers in their original forms without converting to a common format, leading to comparison errors. For example, they might try to directly compare \(0.73 \times 10^{-5}\) with \(\frac{8}{100,000}\) without recognizing both need to be in the same format for accurate comparison.
1. Decimal placement errors during conversion
When converting \(0.73 \times 10^{-5}\), students frequently make mistakes in moving the decimal point. They might move it only 4 places (getting 0.0000073) or 6 places (getting 0.000000073) instead of exactly 5 places to get 0.00000073.
2. Arithmetic errors in basic division and addition
Students may incorrectly calculate \(8 \div 100,000\) as 0.0008 instead of 0.00008, or make errors when adding \(0.00001 + 0.0001\), potentially getting 0.0002 instead of 0.00011 due to misaligning decimal places.
3. Comparison errors with small decimals
When comparing numbers like 0.00008 and 0.000055, students may focus only on the digits 8 and 55, incorrectly concluding that 0.000055 > 0.00008 because 55 > 8, rather than carefully checking the decimal place values.
1. Reversing the inequality order
After correctly determining the relative sizes, students may reverse the order when writing the final inequality. For example, if they correctly identify that a is smallest and c is largest, they might write \(c < b < d < a\) instead of \(a < d < b < c\).
2. Mismatching variables to their calculated values
Students may correctly calculate all decimal values and determine the right order, but then incorrectly map which variable corresponds to which value when selecting from the answer choices, leading to choosing an option like D (\(d < a < b < c\)) instead of the correct E (\(a < d < b < c\)).