What is the tenths digit of the quotient when 0.35 is divided by 0.004?

1. Translate the problem requirements: Convert "thirty-five hundredths" and "four thousandths" into decimal numbers, then find what digit appears in the tenths place of the quotient
2. Convert word forms to decimal notation: Transform the decimal descriptions into standard decimal numbers using place value understanding
3. Set up and solve the division: Divide the two decimal numbers to get the quotient
4. Identify the tenths digit: Locate the first digit to the right of the decimal point in the quotient

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what this problem is asking us to do. We need to:

• Take "thirty-five hundredths" and convert it to a decimal number
• Take "four thousandths" and convert it to a decimal number
• Divide the first number by the second number
• Look at the result and find what digit appears in the tenths place (the first position after the decimal point)

This is really just about understanding decimal place values and doing a division problem.

Process Skill: TRANSLATE - Converting the word descriptions into mathematical operations we can work with

2. Convert word forms to decimal notation

Now let's convert these word descriptions into actual decimal numbers using our understanding of place values.

For "thirty-five hundredths":

• The word "hundredths" tells us we're working with the second decimal place (\(0.01\))
• We have thirty-five of these hundredths
• So thirty-five hundredths = \(35 \times 0.01 = 0.35\)

For "four thousandths":

• The word "thousandths" tells us we're working with the third decimal place (\(0.001\))
• We have four of these thousandths
• So four thousandths = \(4 \times 0.001 = 0.004\)

Our problem becomes: What is the tenths digit when \(0.35\) is divided by \(0.004\)?

3. Set up and solve the division

Now we need to calculate \(0.35 \div 0.004\).

To make this division easier, let's eliminate the decimals by thinking about what we're really doing:

• \(0.35 \div 0.004\) is the same as asking "how many groups of \(0.004\) fit into \(0.35\)?"
• We can multiply both numbers by \(1000\) to eliminate decimals: \((0.35 \times 1000) \div (0.004 \times 1000) = 350 \div 4\)
• Now we can do simple division: \(350 \div 4 = 87.5\)

So our quotient is \(87.5\).

4. Identify the tenths digit

Looking at our quotient \(87.5\), we need to identify the tenths digit.

In the number \(87.5\):

• \(8\) is in the tens place
• \(7\) is in the ones place
• The decimal point separates whole numbers from decimal parts
• \(5\) is in the tenths place (the first position to the right of the decimal point)

Therefore, the tenths digit of the quotient is \(5\).

Final Answer

The tenths digit of the quotient when thirty-five hundredths is divided by four thousandths is \(5\).

This matches answer choice E.

Common Faltering Points

Errors while devising the approach

• Misinterpreting decimal place values: Students often confuse "hundredths" with "hundreds" or "thousandths" with "thousands", leading them to write \(35.00\) instead of \(0.35\), or \(4.000\) instead of \(0.004\). This fundamental misunderstanding of place value terminology derails the entire problem from the start.
• Confusing the division order: Students may set up the division as \(0.004 \div 0.35\) instead of \(0.35 \div 0.004\), misreading which number should be the dividend and which should be the divisor in the phrase "thirty-five hundredths is divided by four thousandths".
• Misunderstanding what "tenths digit" means: Students may think the question is asking for the tens digit (the second digit to the left of the decimal) rather than the tenths digit (the first digit to the right of the decimal point).

Errors while executing the approach

• Arithmetic errors in the division: When calculating \(350 \div 4\), students might make basic division mistakes, getting \(85.5\) or \(87.25\) instead of the correct \(87.5\), which would lead to identifying the wrong tenths digit.
• Incorrectly handling decimal elimination: Students may multiply both numbers by different powers of \(10\) (like multiplying \(0.35\) by \(100\) but \(0.004\) by \(1000\)), which changes the value of the quotient and leads to an incorrect result.

Errors while selecting the answer

• Identifying the wrong digit position: Even after correctly calculating \(87.5\), students may select the ones digit (\(7\)) or accidentally select a different position, confusing themselves about which digit is actually in the tenths place.