Problem 46 - Prealgebra

Question

Simplify the given expression by first converting the fraction into a terminating decimal. \(\frac{3}{2}-2.73\)

Step-by-Step Solution

Answer

The simplified result is -1.23.

Step 1: Convert the Fraction to a Decimal

To simplify the expression, start by converting the fraction \(\frac{3}{2}\) into a decimal. To do this, divide the numerator (3) by the denominator (2). \[\frac{3}{2} = 1.5\] So, \(\frac{3}{2}\) can be written as 1.5 in decimal form.

Step 2: Subtract the Decimal

Now, subtract 2.73 from 1.5. First, align the numbers by their decimal points and then perform the subtraction. \[1.5 - 2.73 = -1.23\] Use borrowing as needed for each place value to complete the subtraction.

Key Concepts

Fraction to Decimal Conversion

Converting a fraction to a decimal is a crucial skill in mathematics, especially when simplifying expressions that involve both fractions and decimals. This process helps us transition from fractional notation to a more easily manipulated decimal form.
To convert a fraction like \(\frac{3}{2}\) to a decimal, we need to perform division. Specifically, divide the numerator (the top number of the fraction) by the denominator (the bottom number).
In the case of \(\frac{3}{2}\), divide 3 by 2:

• 3 divided by 2 results in 1 with a remainder of 1.


• Continue the division by adding a decimal point and zeros to the dividend. In this instance, 3.0 divided by 2 results in 1.5.

Hence, \(\frac{3}{2}\) is equal to 1.5 in decimal form. This is a simple process that turns fractions into decimals, making arithmetic operations more straightforward.

Terminating Decimals

Terminating decimals are decimals that have a limited number of digits following the decimal point. They are important in arithmetic because they provide precise values without an infinitely repeating sequence. For instance, when we converted \(\frac{3}{2}\) to a decimal, the result was 1.5. This decimal ends after a certain point, making it a terminating decimal.
Terminating decimals are a type of rational number and any fraction whose denominator can be expressed as a power of 10 will result in a terminating decimal.
Consider a few examples:

• \(\frac{1}{4} = 0.25\)


• \(\frac{5}{8} = 0.625\)


• \(\frac{1}{5} = 0.2\)

When performing calculations, terminating decimals are easier to work with given they do not require repeating sequences or approximations.

Borrowing in Subtraction

Borrowing, or regrouping, is an essential strategy when subtracting numbers, especially when dealing with decimals. This technique is used when a particular digit in the minuend is smaller than the digit in the same place value column of the subtrahend.
For the expression \(1.5 - 2.73\), borrowing becomes necessary during subtraction. Let's understand how to perform this calculation step by step:

• Align the decimal points of both numbers: 1.50 and 2.73. Add an extra zero to the minuend (1.5) to make it \(1.50\), so it has the same decimal places as 2.73.


• Start subtracting from the right. Here, you can't subtract 3 from 0 in the hundredths place, so you must "borrow." Transform the 5 in the tenths place to 4, and turn the 0 in the hundredths into 10.


• Subtract 3 from 10, leaving 7 in the hundredths place. Move to the tenths place: 4 minus 7 is not possible, thus borrow 1 from the whole number 1, leaving 0. Now, 14 minus 7 gives 7 in the tenths place.


• Lastly, 0 cannot subtract 2, resulting in a negative whole number, \(-1\) as the answer.

The borrowed method ensures that each digit in the subtrahend finds a corresponding digit in the minuend greater than itself, enabling clear and correct operations.