Understanding Number Analogies and Fraction Conversion

This question presents a number analogy problem. In a number analogy, two numbers are related in a specific way, and you need to find a fourth number that is related to a third number in the exact same way. The structure is typically presented as A : B :: C : D, where the relationship between A and B is the same as the relationship between C and D.

In this specific problem, we have the analogy:

\( 2/5 : 0.4 : : 1/200 : ? \)

We need to determine the relationship between \( 2/5 \) and \( 0.4 \) and then apply that same relationship to \( 1/200 \) to find the missing number.

Analyzing the Relationship Between the First Pair

Let's look at the first pair of numbers: \( 2/5 \) and \( 0.4 \).

\( 2/5 \) is a fraction, and \( 0.4 \) is a decimal. Let's try converting the fraction \( 2/5 \) into its decimal equivalent.

To convert a fraction to a decimal, we divide the numerator by the denominator.

\( \frac{2}{5} = 2 \div 5 \)

Performing the division:

• \( 2 \div 5 = 0 \) with a remainder of \( 2 \).
• Place a decimal point after the \( 0 \) in the quotient and add a zero to the dividend, making it \( 2.0 \).
• Now divide \( 20 \) by \( 5 \). \( 20 \div 5 = 4 \).

So, \( \frac{2}{5} = 0.4 \).

The relationship between the first pair is the conversion of the fraction \( 2/5 \) to its decimal form \( 0.4 \).

Applying the Relationship to the Third Number

Now, we apply the same relationship to the third number, which is the fraction \( 1/200 \). We need to convert \( 1/200 \) into its decimal equivalent.

\( \frac{1}{200} = 1 \div 200 \)

Let's perform the division \( 1 \div 200 \):

• \( 1 \div 200 = 0 \) with a remainder of \( 1 \).
• Place a decimal point after the \( 0 \) in the quotient and add zeros to the dividend. Start with \( 1.0 \).
• \( 10 \div 200 = 0 \) with a remainder of \( 10 \). Add another zero, making it \( 1.00 \).
• \( 100 \div 200 = 0 \) with a remainder of \( 100 \). Add another zero, making it \( 1.000 \).
• \( 1000 \div 200 \). \( 200 \times 5 = 1000 \). So, \( 1000 \div 200 = 5 \).

Thus, \( \frac{1}{200} = 0.005 \).

Comparing with the Options

The decimal equivalent of \( 1/200 \) is \( 0.005 \). Let's check the given options:

1. 0.005
2. 0.05
3. 0.2
4. 0.002

The calculated value \( 0.005 \) matches option 1.

Therefore, the number that is related to \( 1/200 \) in the same way as \( 0.4 \) is related to \( 2/5 \) is \( 0.005 \).

Revision Table: Key Concepts

Additional Information: Decimal Place Values

Understanding decimal place values is crucial for converting fractions to decimals accurately, especially when dealing with denominators that result in decimal places beyond the tenths.

• The first digit after the decimal point is the tenths place (e.g., \( 0.4 \) means \( 4/10 \)).
• The second digit is the hundredths place (e.g., \( 0.05 \) means \( 5/100 \)).
• The third digit is the thousandths place (e.g., \( 0.005 \) means \( 5/1000 \)).

When converting \( 1/200 \) to a decimal, we get \( 0.005 \). This means \( 5 \) in the thousandths place, which is equivalent to \( 5/1000 \). Let's check if \( 1/200 \) is equivalent to \( 5/1000 \):

\( \frac{5}{1000} = \frac{5 \div 5}{1000 \div 5} = \frac{1}{200} \)

This confirms our conversion is correct and reinforces the understanding of decimal place values.