Q. \( \frac{1}{3} \times \frac{1}{3} \).

Q: What is \tfrac{1}{3}\times\tfrac{1}{3} ?

Multiply numerators and denominators: \tfrac{1\times1}{3\times3} = \tfrac{1}{9} .

Q: Why do we multiply numerators and denominators?

Because \tfrac{a}{b}\times\tfrac{c}{d}=\tfrac{ac}{bd} ; multiplying ratios preserves the proportion of parts of a whole.

Q: Is \tfrac{1}{3}\times\tfrac{1}{3} the same as \left(\tfrac{1}{3}\right)^2 ?

Yes. Squaring a fraction multiplies it by itself: \left(\tfrac{1}{3}\right)^2=\tfrac{1}{9} .

Q: What is \tfrac{1}{9} as a decimal?

\tfrac{1}{9} equals 0.111... (repeating), written 0.\overline{1} .

Q: What common mistakes should I avoid?

Don’t add when you should multiply ( \tfrac{1}{3}+\tfrac{1}{3}=\tfrac{2}{3} ). Avoid dividing numerator by denominator unless reducing factors correctly.

Q: How does this extend to more fractions?

Multiply all numerators and all denominators: \tfrac{1}{3}\times\tfrac{1}{3}\times\tfrac{1}{3}=\tfrac{1}{27} .

Answer

Multiply numerators and denominators: \frac{1}{3}\times\frac{1}{3}=\frac{1\cdot1}{3\cdot3}=\frac{1}{9}

Detailed Explanation

Problem

Compute the product \( \frac{1}{3} \times \frac{1}{3} \).

Step-by-step explanation

Write the multiplication of two fractions. The expression is

\[ \frac{1}{3} \times \frac{1}{3} \]
Recall the rule for multiplying fractions: to multiply two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\), multiply the numerators together and multiply the denominators together. In formula form,

\[ \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \]

We apply this rule with \(a = 1\), \(b = 3\), \(c = 1\), \(d = 3\).
Multiply the numerators and multiply the denominators separately:

\[ \frac{1}{3} \times \frac{1}{3} = \frac{1 \times 1}{3 \times 3} \]
Compute the products in numerator and denominator:

\[ 1 \times 1 = 1 \quad\text{and}\quad 3 \times 3 = 9 \]

So the fraction becomes

\[ \frac{1}{9} \]
Check for simplification: the greatest common divisor of 1 and 9 is 1, so the fraction \(\frac{1}{9}\) is already in simplest form.

Final answer

\[ \frac{1}{3} \times \frac{1}{3} = \frac{1}{9} \]

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Arithmetic FAQs

What is \( \tfrac{1}{3}\times\tfrac{1}{3} \)?

Multiply numerators and denominators: \( \tfrac{1\times1}{3\times3} = \tfrac{1}{9} \).

Why do we multiply numerators and denominators?

Because \( \tfrac{a}{b}\times\tfrac{c}{d}=\tfrac{ac}{bd} \); multiplying ratios preserves the proportion of parts of a whole.

Can I simplify before multiplying?

Yes. Cross-cancel any common factors between a numerator and an opposite denominator to keep numbers small. Here there’s nothing to cancel, so multiply directly.

How can I visualize \( \tfrac{1}{3}\times\tfrac{1}{3} \)?

Take one third of a whole, then take one third of that part. You now have \( \tfrac{1}{9} \) of the original whole.

Is \( \tfrac{1}{3}\times\tfrac{1}{3} \) the same as \( \left(\tfrac{1}{3}\right)^2 \)?

Yes. Squaring a fraction multiplies it by itself: \( \left(\tfrac{1}{3}\right)^2=\tfrac{1}{9} \).

What is \( \tfrac{1}{9} \) as a decimal?

\( \tfrac{1}{9} \) equals 0.111... (repeating), written \( 0.\overline{1} \).

What common mistakes should I avoid?

Don’t add when you should multiply ( \( \tfrac{1}{3}+\tfrac{1}{3}=\tfrac{2}{3} \) ). Avoid dividing numerator by denominator unless reducing factors correctly.

How does this extend to more fractions?

Multiply all numerators and all denominators: \( \tfrac{1}{3}\times\tfrac{1}{3}\times\tfrac{1}{3}=\tfrac{1}{27} \).

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