Question

Apply the principles of borrowing, and subtract the following: \(2-\frac{10}{21}=\) ______

Short answer

The answer is \(\frac{32}{21}\).

Step-by-step solution

Understand the Problem

We have the expression \(2 - \frac{10}{21}\). To solve it, we need to subtract a fraction from a whole number. This may involve converting the whole number into a fraction for easier subtraction.

Convert Whole Number to Fraction

Convert the whole number \(2\) into a fraction with a denominator of 21. This can be done by expressing 2 as \(\frac{42}{21}\) because \(2 \times 21 = 42\).

Set Up the Fraction Subtraction

Now, our problem is \(\frac{42}{21} - \frac{10}{21}\). Both fractions have the same denominator, so we can subtract the numerators directly.

Subtract the Numerators

Subtract the numerators: \(42 - 10 = 32\). This leaves us with \(\frac{32}{21}\) as the result of the subtraction.

Simplify the Fraction

Since \(\frac{32}{21}\) is already in its simplest form (as 32 and 21 have no common factors other than 1), no further simplification is needed.

Key concepts

Fraction Subtraction

Subtracting fractions can sometimes be confusing, but breaking it down step by step makes it easier. First, ensure that the fractions involved have the same denominator. This common denominator allows you to focus on subtracting just the numerators. In our example, both \( \frac{42}{21} \) and \( \frac{10}{21} \) share the denominator 21, making direct subtraction of the numerators possible.

• Align your fractions under a common denominator.
• Subtract the numerators while maintaining the denominator.
• Simplify if possible, although in this example, the fraction is already simplified.

Borrowing Principles

Borrowing in mathematics, especially with fractions, involves breaking down a whole number to make subtraction smoother. When dealing with a whole number like 2 and subtracting a fraction, you might convert that whole number into a fraction equivalent with a denominator matching the fraction you’re working with.

• Convert the whole number into an equivalent fraction of the same denominator.
• This step is crucial because you cannot directly subtract a proper fraction from a whole number without equalizing denominators first.
• The idea is akin to borrowing in subtraction, where you adjust numbers to make operations possible.

Whole Numbers as Fractions

Turning a whole number into a fraction is a handy trick in algebra and arithmetic. This method ensures that numbers are compatible for subtraction with fractions. In our problem, the whole number 2 was converted to \( \frac{42}{21} \). This conversion used multiples of the denominator to maintain equality.

• Multiply the whole number by the denominator of the fraction you are working with.
• Express the whole number as a fraction using this multiplication.
• By converting, you ensure the whole number stands compatible for direct subtraction.

Numerator Subtraction

Subtraction of numerators is straightforward once the fractions align in terms of their denominators. It involves subtracting the top numbers of the fractions. In our solved problem, once we had \( \frac{42}{21} \) and \( \frac{10}{21} \), the next step was simply to subtract 10 from 42.

• Ensure the denominators are the same before subtracting numerators.
• Perform the subtraction: here \( 42 - 10 = 32 \).
• Write your result over the common denominator.

Subtracting numerators is uncomplicated but depends heavily on having matched denominators.