Area is the measure of the amount of space inside the boundary of a two-dimensional (2D) flat shape or the surface of a 3D object. It's expressed in square units, such as square meters (m²), square feet (ft²), square inches (in²), etc.

This question covers phrasings like: "how to find area", "how to calculate area", "how do you find area", "what is the formula for area", "how to determine area", "how to find the area of a shape/figure", "how to figure out area", "how to get area", "how do i find area".

To find the area of a shape:

1. Identify the shape: Is it a rectangle, square, circle, triangle, or a more complex figure?
2. Know the formula: Each shape has a specific formula to calculate its area. There isn't one single formula for "area" that applies to all shapes.
3. Measure dimensions: You'll need to measure lengths, widths, heights, bases, or radii, depending on the shape and its formula. Ensure all measurements are in the same unit.
4. Apply the formula: Substitute your measurements into the correct formula and perform the calculation.
5. State the units: The result should be in square units (e.g., if dimensions were in cm, area is in cm²).

For complex shapes, you might need to break them down into simpler shapes, calculate their individual areas, and then add or subtract those areas as needed.

A rectangle is a four-sided shape with four right angles (90°), where opposite sides are equal in length.

This covers: "how do you find the area of a rectangle", "how to find area of a rectangle", "what is the area of a rectangle", "how to calculate the area of a rectangle", "a of rectangle", "how to find area of rectangle", "how to get the area of a rectangle".

Formula:

Area (A) = Length (l) × Width (w)

Or sometimes expressed as:

A = l ⋅ w

Steps:

1. Measure the length (the longer side) of the rectangle.
2. Measure the width (the shorter side) of the rectangle. Ensure both measurements are in the same unit.
3. Multiply the length by the width.

Example: If a rectangle has a length of 10 cm and a width of 5 cm, its area is:

A = 10 cm × 5 cm = 50 cm²

A square is a special type of rectangle where all four sides are of equal length, and all four angles are right angles (90°).

This covers: "how to find the area of a square", "how do you find the area of a square", "what is the area of a square", "how to get the area of a square", "how to find area of square".

Formula:

Area (A) = Side (s) × Side (s) = s²

Steps:

1. Measure the length of one side of the square. Since all sides are equal, you only need one measurement.
2. Multiply the side length by itself (square the side length).

Example: If a square has a side length of 7 inches, its area is:

A = 7 in × 7 in = 49 in²

A triangle is a three-sided polygon.

This covers: "how to find area of a triangle", "how to calculate the area of a triangle".

Formula:

Area (A) = ¹⁄₂ × Base (b) × Height (h)

Where:

• Base (b): Any one side of the triangle.
• Height (h): The perpendicular distance from the base to the opposite vertex (corner).

Steps:

1. Identify the base of the triangle.
2. Determine the height corresponding to that base. The height must be perpendicular (at a 90° angle) to the chosen base.
3. Multiply the base by the height.
4. Divide the result by 2 (or multiply by ¹⁄₂).

Example: If a triangle has a base of 10 meters and a corresponding height of 6 meters, its area is:

A = ¹⁄₂ × 10 m × 6 m = ¹⁄₂ × 60 m² = 30 m²

Calculating square footage (sq ft or ft²) is finding the area of a space in units of square feet. This is commonly used for rooms, houses, or land.

This covers: "how do you calculate square footage", "how to calculate sq ft", "how calculate sq ft".

For a rectangular or square area:

1. Measure the length of the area in feet.
2. Measure the width of the area in feet.
3. Multiply the length by the width. The result is the area in square feet.

Square Footage = Length (ft) × Width (ft)

Example: A room is 12 feet long and 10 feet wide.

Area = 12 ft × 10 ft = 120 ft²

For non-rectangular areas: Break the area into simpler shapes (rectangles, triangles), calculate their individual square footage, and then add them together.

Converting to Square Yards (sq yd or yd²):

There are 3 feet in 1 yard, so there are 3 ft × 3 ft = 9 square feet in 1 square yard.

1 yd² = 9 ft²

To convert square feet to square yards, divide the square footage by 9.

Area (yd²) = Area (ft²) ÷ 9

Example: If an area is 180 ft².

Area in square yards = 180 ft² ÷ 9 = 20 yd²

This also helps with "how to figure out square yards".

Acreage is a measure of land area, commonly used in countries like the United States and the United Kingdom. One acre is defined as 43,560 square feet.

1 acre = 43,560 ft²

Steps to calculate acreage:

1. Calculate the area in square feet: Use the appropriate methods (length × width for rectangles, or more complex methods for irregular shapes) to find the total square footage of the land.
2. Convert square feet to acres: Divide the total square footage by 43,560.

Acreage = Total Square Feet ÷ 43,560

Example: A plot of land is rectangular, measuring 300 feet long by 200 feet wide.

1. Area in square feet = 300 ft × 200 ft = 60,000 ft²
2. Acreage = 60,000 ft² ÷ 43,560 ft²/acre ≈ 1.377 acres

If measurements are in other units (like meters), first convert them to feet or calculate the area in square meters and then convert square meters to acres (1 acre ≈ 4046.86 m²).

A fraction represents a part of a whole or, more generally, any number of equal parts. It is written as one number over another, separated by a line.

This covers: "what is a fraction", "a fraction".

^Numerator⁄_Denominator

• The Numerator (top number) tells you how many equal parts you have.
• The Denominator (bottom number) tells you how many equal parts the whole is divided into. The denominator cannot be zero.

Example: The fraction ³⁄₄ (three-fourths) means you have 3 parts, and the whole was divided into 4 equal parts.

The question "what is 2/3" refers to the fraction ²⁄₃, meaning two parts out of a whole that is divided into three equal parts.

Fractions can also represent division: ^a⁄_b is equivalent to a ÷ b.

An improper fraction is a fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number).

This means the fraction represents a value that is one whole or more than one whole.

Examples:

• ⁵⁄₄ (five-fourths) - Numerator (5) is greater than denominator (4).
• ⁷⁄₇ (seven-sevenths) - Numerator (7) is equal to denominator (7). This equals 1.
• ¹⁰⁄₃ (ten-thirds) - Numerator (10) is greater than denominator (3).

Improper fractions can be converted into mixed numbers (a whole number and a proper fraction).

A mixed number (also called a mixed fraction) is a number consisting of a whole number and a proper fraction combined.

This covers: "what is a mixed number", "what's a mixed number".

It represents a value greater than one.

Format: Whole Number ^Numerator⁄_Denominator

Examples:

• 1 ¹⁄₂ (one and one-half)
• 3 ²⁄₅ (three and two-fifths)
• 7 ³⁄₄ (seven and three-fourths)

A mixed number can always be converted into an improper fraction, and vice-versa (if the improper fraction is greater than 1).

For example, 1 ¹⁄₂ means 1 whole + ¹⁄₂.

Simplifying (or reducing) a fraction means to express it in its simplest form, where the numerator and the denominator have no common factors other than 1. This means finding an equivalent fraction with the smallest possible whole numbers for the numerator and denominator.

This covers: "how to simplify fractions", "how do you simplify fractions", "how to simplify a fraction", "how to reduce fractions", "how do you reduce a fraction".

Steps:

1. Find the Greatest Common Divisor (GCD): Find the largest number that divides both the numerator and the denominator evenly. This is also known as the Greatest Common Factor (GCF).
2. Divide: Divide both the numerator and the denominator by their GCD.

Example 1: Simplify ¹²⁄₁₈

1. Factors of 12: 1, 2, 3, 4, 6, 12
2. Factors of 18: 1, 2, 3, 6, 9, 18
3. The GCD of 12 and 18 is 6.
4. Divide both by 6:

   ^{(12 ÷ 6)}⁄_{(18 ÷ 6)} = ²⁄₃

So, ¹²⁄₁₈ simplified is ²⁄₃.

Alternative Method (Prime Factorization):

1. Write the prime factorization of the numerator and the denominator.
2. Cancel out common prime factors.
3. Multiply the remaining factors in the numerator and denominator.

Example 2: Simplify ²⁴⁄₃₆

1. Prime factorization of 24: 2 × 2 × 2 × 3
2. Prime factorization of 36: 2 × 2 × 3 × 3

3. ²⁴⁄₃₆ = ^{(2 × 2 × 2 × 3)}⁄_{(2 × 2 × 3 × 3)}

4. Cancel common factors (two 2s and one 3):

   = ^{(✘2 × ✘2 × 2 × ✘3)}⁄_{(✘2 × ✘2 × ✘3 × 3)} = ²⁄₃

   (Using ✘ to represent cancellation for visual aid: 2 would be another option if styled correctly)

To convert a mixed number (e.g., W ⁿ⁄_d, where W is the whole number, n is the numerator, and d is the denominator) into an improper fraction:

This covers: "how do i convert mixed numbers into improper fractions", "how to turn a mixed number into an improper fraction", "how do you convert mixed fractions into improper fractions", "how to change mixed numbers to improper fractions".

Formula / Steps:

1. Multiply the whole number by the denominator of the fraction.
2. Add the result to the numerator of the fraction.
3. Keep the original denominator. This new sum is the numerator of the improper fraction, and the denominator remains the same.

Improper Fraction = ^{( (Whole Number × Denominator) + Numerator )}⁄_Denominator

Example: Convert 3 ²⁄₅ to an improper fraction.

1. Multiply whole number by denominator: 3 × 5 = 15
2. Add the result to the numerator: 15 + 2 = 17
3. Keep the original denominator: 5

So, 3 ²⁄₅ = ¹⁷⁄₅

To convert an improper fraction (where the numerator is greater than or equal to the denominator) into a mixed number:

This covers: "how to convert improper fractions to mixed numbers", "how to change an improper fraction to a mixed number", "how to turn an improper fraction into a mixed number", "how do i convert improper fractions to mixed numbers".

Steps:

1. Divide the numerator by the denominator.
2. The whole number part of the mixed number is the quotient (the result of the division).
3. The numerator of the fractional part is the remainder of the division.
4. The denominator of the fractional part is the original denominator.

Example: Convert ¹⁷⁄₅ to a mixed number.

1. Divide 17 by 5:

   17 ÷ 5 = 3 with a remainder of 2.

2. Whole number part: 3
3. Numerator of fraction: 2 (the remainder)
4. Denominator of fraction: 5 (the original denominator)

So, ¹⁷⁄₅ = 3 ²⁄₅

If the remainder is 0, the improper fraction simplifies to a whole number (e.g., ¹⁰⁄₅ = 2).

To multiply a fraction by a whole number (or a whole number by a fraction):

This covers: "how to multiply fractions with whole numbers", "how do you multiply a fraction by a whole number", "how to multiply fractions and whole numbers", "how to multiply a fraction with whole number", "how to multiply fractions by whole numbers", "how to multiply whole numbers and fractions", "how to multiply a fraction times a whole number".

Steps:

1. Convert the whole number to a fraction: Write the whole number as a fraction with a denominator of 1. (e.g., 5 becomes ⁵⁄₁).
2. Multiply the numerators: Multiply the numerator of the first fraction by the numerator of the second fraction.
3. Multiply the denominators: Multiply the denominator of the first fraction by the denominator of the second fraction.
4. Simplify: If necessary, simplify the resulting fraction.

(^a⁄_b) × c = (^a⁄_b) × (^c⁄₁) = ^{(a × c)}⁄_{(b × 1)} = ^{(a × c)}⁄_b

Essentially, you just multiply the whole number by the numerator of the fraction, and keep the denominator the same.

Example: Multiply ²⁄₃ by 5.

²⁄₃ × 5 = ²⁄₃ × ⁵⁄₁ = ^{(2 × 5)}⁄_{(3 × 1)} = ¹⁰⁄₃

As a mixed number, ¹⁰⁄₃ is 3 ¹⁄₃.

To multiply mixed numbers (or a mixed number by a fraction):

This covers: "how to multiply mixed fractions", "how to multiply mixed numbers", "how do you times mixed fractions", "how to multiplying mixed numbers", "how do you multiply mixed numbers", "how do you multiply mixed numbers by fractions", "how to multiply fractions with mixed numbers".

Steps:

1. Convert mixed numbers to improper fractions: Change each mixed number into an improper fraction. (See "How to Convert Mixed Numbers to Improper Fractions".)
2. Multiply the numerators: Multiply the numerators of the improper fractions together.
3. Multiply the denominators: Multiply the denominators of the improper fractions together.
4. Simplify: If the result is an improper fraction, convert it back to a mixed number and/or simplify it.

Example 1: Multiply two mixed numbers: 1 ¹⁄₂ × 2 ¹⁄₃

1. Convert to improper fractions:

   1 ¹⁄₂ = ^{( (1 × 2) + 1 )}⁄₂ = ³⁄₂

   2 ¹⁄₃ = ^{( (2 × 3) + 1 )}⁄₃ = ⁷⁄₃

2. Multiply the improper fractions:

   ³⁄₂ × ⁷⁄₃ = ^{(3 × 7)}⁄_{(2 × 3)} = ²¹⁄₆

3. Simplify and convert to mixed number:

   ²¹⁄₆ can be simplified by dividing numerator and denominator by 3: ⁷⁄₂.

   Convert ⁷⁄₂ to a mixed number: 7 ÷ 2 = 3 with a remainder of 1. So, ⁷⁄₂ = 3 ¹⁄₂.

Example 2: Multiply a mixed number by a fraction: 2 ¹⁄₄ × ²⁄₃

1. Convert mixed number: 2 ¹⁄₄ = ^{( (2 × 4) + 1 )}⁄₄ = ⁹⁄₄
2. Multiply: ⁹⁄₄ × ²⁄₃ = ^{(9 × 2)}⁄_{(4 × 3)} = ¹⁸⁄₁₂
3. Simplify: GCD of 18 and 12 is 6.

   ^{(18 ÷ 6)}⁄_{(12 ÷ 6)} = ³⁄₂

   As a mixed number: 1 ¹⁄₂.

To divide a whole number by a fraction:

This covers: "how do you divide a whole number by a fraction".

Steps (Keep, Change, Flip):

1. Convert the whole number to a fraction: Write the whole number as a fraction with a denominator of 1 (e.g., 5 becomes ⁵⁄₁). This is the dividend.
2. Find the reciprocal of the divisor fraction: "Flip" the fraction you are dividing by (the divisor). Swap its numerator and denominator. (e.g., the reciprocal of ^a⁄_b is ^b⁄_a).
3. Change division to multiplication: Change the division sign to a multiplication sign.
4. Multiply: Multiply the first fraction (the whole number as a fraction) by the reciprocal of the second fraction.
5. Simplify: Simplify the result if necessary.

c ÷ (^a⁄_b) = (^c⁄₁) × (^b⁄_a) = ^{(c × b)}⁄_{(1 × a)} = ^{(c × b)}⁄_a

Example: Divide 5 by ²⁄₃.

1. Whole number as fraction: 5 = ⁵⁄₁
2. Reciprocal of ²⁄₃ is ³⁄₂.
3. Change to multiplication: ⁵⁄₁ × ³⁄₂
4. Multiply: ^{(5 × 3)}⁄_{(1 × 2)} = ¹⁵⁄₂
5. Simplify (convert to mixed number): ¹⁵⁄₂ = 7 ¹⁄₂.

To divide a fraction by a whole number:

Steps (Keep, Change, Flip):

1. Keep the first fraction (the dividend) as it is.
2. Convert the whole number (the divisor) to a fraction: Write the whole number as a fraction with a denominator of 1 (e.g., 5 becomes ⁵⁄₁).
3. Find the reciprocal of the whole number fraction: "Flip" the fraction representing the whole number. (e.g., the reciprocal of ^c⁄₁ is ¹⁄_c).
4. Change division to multiplication: Change the division sign to a multiplication sign.
5. Multiply: Multiply the first fraction by the reciprocal of the whole number fraction.
6. Simplify: Simplify the result if necessary.

(^a⁄_b) ÷ c = (^a⁄_b) ÷ (^c⁄₁) = (^a⁄_b) × (¹⁄_c) = ^{(a × 1)}⁄_{(b × c)} = ^a⁄_{(b × c)}

Essentially, you multiply the denominator of the fraction by the whole number, and keep the numerator the same.

Example: Divide ³⁄₄ by 2.

1. Keep ³⁄₄.
2. Whole number as fraction: 2 = ²⁄₁.
3. Reciprocal of ²⁄₁ is ¹⁄₂.
4. Change to multiplication: ³⁄₄ × ¹⁄₂
5. Multiply: ^{(3 × 1)}⁄_{(4 × 2)} = ³⁄₈

Dividing fractions involves multiplying by the reciprocal of the second fraction (the divisor). This is often remembered by the mnemonic "Keep, Change, Flip".

This covers: "how to divide fractions with whole numbers", "how to divide fractions and whole numbers". (Specific cases with whole numbers are detailed in separate FAQs but follow this general rule).

Steps to divide fraction ^a⁄_b by fraction ^c⁄_d:

(^a⁄_b) ÷ (^c⁄_d)

1. Keep: Keep the first fraction (the dividend) the same: ^a⁄_b.
2. Change: Change the division sign (÷) to a multiplication sign (×).
3. Flip: Take the reciprocal of the second fraction (the divisor). Flip ^c⁄_d to get ^d⁄_c.
4. Multiply: Multiply the numerators together and the denominators together.

   (^a⁄_b) × (^d⁄_c) = ^{(a × d)}⁄_{(b × c)}

5. Simplify: Simplify the resulting fraction if possible.

Example: Divide ²⁄₃ by ⁴⁄₅.

(²⁄₃) ÷ (⁴⁄₅) = (²⁄₃) × (⁵⁄₄) = ^{(2 × 5)}⁄_{(3 × 4)} = ¹⁰⁄₁₂

Simplify ¹⁰⁄₁₂ by dividing numerator and denominator by their GCD (2):

^{(10 ÷ 2)}⁄_{(12 ÷ 2)} = ⁵⁄₆

When whole numbers are involved: First convert the whole number to a fraction (e.g., N becomes ^N⁄₁) and then apply the "Keep, Change, Flip" rule.

To divide mixed numbers, or a mixed number by a fraction (or vice versa):

This covers: "how to divide mixed numbers", "how to divide mixed fractions", "how to divide fractions with mixed numbers".

Steps:

1. Convert to Improper Fractions: Convert all mixed numbers into improper fractions. (If one of the numbers is already a proper or improper fraction, leave it as is).
2. Apply "Keep, Change, Flip":
   • Keep the first improper fraction (the dividend).
   • Change the division sign to a multiplication sign.
   • Flip the second improper fraction (the divisor) to get its reciprocal.
3. Multiply: Multiply the numerators together and the denominators together.
4. Simplify: Simplify the resulting fraction. If it's an improper fraction, you may want to convert it back to a mixed number.

Example: Divide 2 ¹⁄₂ by 1 ¹⁄₄

1. Convert to improper fractions:

   2 ¹⁄₂ = ^{( (2 × 2) + 1 )}⁄₂ = ⁵⁄₂

   1 ¹⁄₄ = ^{( (1 × 4) + 1 )}⁄₄ = ⁵⁄₄

2. Apply "Keep, Change, Flip":

   (⁵⁄₂) ÷ (⁵⁄₄) = (⁵⁄₂) × (⁴⁄₅)

3. Multiply:

   ^{(5 × 4)}⁄_{(2 × 5)} = ²⁰⁄₁₀

4. Simplify:

   ²⁰⁄₁₀ = 2

There are two common methods to add mixed fractions (mixed numbers):

Method 1: Add Whole Parts and Fractional Parts Separately

1. Add the whole number parts.
2. Add the fractional parts:
   • Find a common denominator for the fractions.
   • Convert the fractions to equivalent fractions with the common denominator.
   • Add the numerators of these equivalent fractions, keeping the common denominator.
3. Combine results: Add the sum of the whole numbers to the sum of the fractions.
4. Simplify: If the fractional part of the sum is an improper fraction, convert it to a mixed number and add its whole part to the total whole number. Simplify the final fractional part if needed.

Example: 2 ¹⁄₄ + 1 ¹⁄₂

1. Add whole numbers: 2 + 1 = 3.
2. Add fractional parts: ¹⁄₄ + ¹⁄₂.
   • Common denominator for 4 and 2 is 4.
   • ¹⁄₂ = ^{(1 × 2)}⁄_{(2 × 2)} = ²⁄₄.
   • So, ¹⁄₄ + ²⁄₄ = ⁽¹⁺²⁾⁄₄ = ³⁄₄.
3. Combine: 3 + ³⁄₄ = 3 ³⁄₄.

Method 2: Convert to Improper Fractions First

1. Convert all mixed numbers to improper fractions.
2. Find a common denominator for the improper fractions.
3. Convert the improper fractions to equivalent fractions with the common denominator.
4. Add the numerators, keeping the common denominator.
5. Convert back: If the result is an improper fraction, convert it back to a mixed number. Simplify if needed.

Example: 2 ¹⁄₄ + 1 ¹⁄₂

1. Convert to improper fractions:

   2 ¹⁄₄ = ⁹⁄₄

   1 ¹⁄₂ = ³⁄₂

2. Common denominator for 4 and 2 is 4. Convert ³⁄₂ to ⁶⁄₄.
3. Add: ⁹⁄₄ + ⁶⁄₄ = ⁽⁹⁺⁶⁾⁄₄ = ¹⁵⁄₄.
4. Convert back: 15 ÷ 4 = 3 with a remainder of 3. So, ¹⁵⁄₄ = 3 ³⁄₄.

This covers: "how to add fractions with whole numbers", "how to add whole numbers and fractions".

Adding a Whole Number and a Fraction:

This is straightforward: you simply combine them to form a mixed number.

Whole Number + ^Numerator⁄_Denominator = Whole Number ^Numerator⁄_Denominator

Example: 3 + ²⁄₅

3 + ²⁄₅ = 3 ²⁄₅

If you need the result as an improper fraction:

1. Convert the whole number to a fraction with the same denominator as the other fraction.

   For 3 + ²⁄₅, convert 3 to fifths: 3 = ^{(3 × 5)}⁄₅ = ¹⁵⁄₅.

2. Add the fractions: ¹⁵⁄₅ + ²⁄₅ = ⁽¹⁵⁺²⁾⁄₅ = ¹⁷⁄₅.

Adding two Fractions (general, for context):

1. Find a common denominator: If the denominators are different, find the least common multiple (LCM) of the denominators. This will be your common denominator.
2. Convert to equivalent fractions: Rewrite each fraction as an equivalent fraction with the common denominator.
3. Add the numerators: Add the numerators of the new fractions. The denominator remains the common denominator.
4. Simplify: Simplify the resulting fraction if possible. If it's an improper fraction, you might convert it to a mixed number.

Example: ¹⁄₃ + ¹⁄₂

1. Common denominator for 3 and 2 is 6.
2. Convert: ¹⁄₃ = ²⁄₆ and ¹⁄₂ = ³⁄₆.
3. Add: ²⁄₆ + ³⁄₆ = ⁽²⁺³⁾⁄₆ = ⁵⁄₆.

This covers: "how to subtract mixed fractions", "how to subtract mixed numbers".

Method 1: Convert to Improper Fractions First (Often Easiest)

1. Convert all mixed numbers to improper fractions.
2. Find a common denominator for the improper fractions.
3. Convert the improper fractions to equivalent fractions with the common denominator.
4. Subtract the numerators, keeping the common denominator.
5. Convert back: If the result is an improper fraction, convert it back to a mixed number. Simplify if needed.

Example: 3 ¹⁄₄ - 1 ¹⁄₂

1. Convert to improper fractions:

   3 ¹⁄₄ = ¹³⁄₄

   1 ¹⁄₂ = ³⁄₂

2. Common denominator is 4. Convert ³⁄₂ to ⁶⁄₄.
3. Subtract: ¹³⁄₄ - ⁶⁄₄ = ^(13-6)⁄₄ = ⁷⁄₄.
4. Convert back: 7 ÷ 4 = 1 with a remainder of 3. So, ⁷⁄₄ = 1 ³⁄₄.

Method 2: Subtract Whole Parts and Fractional Parts (Requires Borrowing if Needed)

1. Subtract the fractional parts:
   • Find a common denominator. Convert fractions.
   • If the first fraction is smaller than the second, you'll need to "borrow" 1 from the whole number part of the first mixed number. Add this borrowed 1 (in the form of ^denominator⁄_denominator) to the first fraction before subtracting.
2. Subtract the whole number parts. (Adjust the first whole number if you borrowed from it).
3. Combine results. Simplify if needed.

Example (with borrowing): 3 ¹⁄₄ - 1 ³⁄₄

1. Fractional parts: ¹⁄₄ - ³⁄₄. We can't subtract 3 from 1 directly.
   • Borrow 1 from the 3 in 3 ¹⁄₄. The 3 becomes 2.
   • The borrowed 1 is ⁴⁄₄. Add to the existing fraction: ¹⁄₄ + ⁴⁄₄ = ⁵⁄₄.
   • Now subtract: ⁵⁄₄ - ³⁄₄ = ²⁄₄. Simplify to ¹⁄₂.
2. Subtract whole numbers (remembering the borrow): 2 - 1 = 1.
3. Combine: 1 ¹⁄₂.

This involves subtracting a fraction from a whole number, or a whole number from a mixed number (which is covered by "Subtract Mixed Fractions" if the mixed number is converted to an improper fraction).

Case 1: Whole Number - Fraction (e.g., 5 - ²⁄₃)

1. Borrow 1 from the whole number: Reduce the whole number by 1.
2. Convert the borrowed 1 into a fraction with the same denominator as the fraction being subtracted. (e.g., if subtracting ²⁄₃, the borrowed 1 becomes ³⁄₃).
3. Subtract the fractions.
4. Combine the new whole number with the resulting fraction.

Example: 5 - ²⁄₃

1. Borrow 1 from 5, so 5 becomes 4.
2. The borrowed 1 is ³⁄₃.
3. Subtract fractions: ³⁄₃ - ²⁄₃ = ¹⁄₃.
4. Combine: 4 ¹⁄₃.

Alternative for Case 1 (using improper fractions):

1. Convert the whole number to an improper fraction with the same denominator as the fraction being subtracted.

   5 = ^{(5 × 3)}⁄₃ = ¹⁵⁄₃

2. Subtract the fractions: ¹⁵⁄₃ - ²⁄₃ = ¹³⁄₃
3. Convert back to mixed number if needed: ¹³⁄₃ = 4 ¹⁄₃

Case 2: Fraction - Whole Number (e.g., ²⁄₃ - 5)

This will result in a negative number if the fraction is proper.

1. Convert the whole number to an improper fraction with the same denominator as the other fraction.

   5 = ¹⁵⁄₃

2. Subtract: ²⁄₃ - ¹⁵⁄₃ = ^{(2 - 15)}⁄₃ = ^-13⁄₃
3. Convert to a negative mixed number if needed: ^-13⁄₃ = -4 ¹⁄₃

To convert a negative decimal like -4.125 to an improper fraction:

1. Handle the negative sign: Keep the negative sign aside for now and apply it to the final improper fraction. Focus on converting 4.125.
2. Separate the whole number and decimal part: 4.125 = 4 + 0.125.
3. Convert the decimal part to a fraction:
   • 0.125 has three decimal places. So, write 125 over 1000 (1 followed by three zeros): ¹²⁵⁄₁₀₀₀.
4. Simplify this fraction:
   • Find the GCD of 125 and 1000. Both are divisible by 125.
   • 125 ÷ 125 = 1
   • 1000 ÷ 125 = 8
   • So, ¹²⁵⁄₁₀₀₀ = ¹⁄₈.
5. Combine the whole number and the simplified fraction: This gives a mixed number: 4 + ¹⁄₈ = 4 ¹⁄₈.
6. Convert the mixed number to an improper fraction:

   4 ¹⁄₈ = ^{( (4 × 8) + 1 )}⁄₈ = ^{(32 + 1)}⁄₈ = ³³⁄₈.

7. Apply the negative sign: Since the original number was -4.125, the improper fraction is -³³⁄₈.

"How to do fractions" is a broad question. It generally refers to understanding and performing operations with fractions. Key aspects include:

• Understanding what fractions represent: Parts of a whole.
• Types of fractions: Proper, improper, mixed numbers.
• Conversions: Between mixed numbers and improper fractions.
• Simplifying fractions: Reducing to lowest terms.
• Comparing fractions: Determining which is larger or smaller.
• Basic operations:
  • Addition and Subtraction: Require a common denominator.
  • Multiplication: Multiply numerators, multiply denominators. Convert mixed numbers to improper first.
  • Division: Multiply by the reciprocal of the divisor ("Keep, Change, Flip"). Convert mixed numbers to improper first.

The other FAQs in this guide cover these operations in detail. This question also implicitly covers "how to fractions".

How to do fractions on a calculator:

• Basic Calculators (like the one provided here): Most basic calculators work primarily with decimals. To perform fraction operations:
  1. Convert fractions to decimals: Divide the numerator by the denominator (e.g., ³⁄₄ = 3 ÷ 4 = 0.75).
  2. Perform the operation using the decimal values.
  3. Interpret the result: The result will be a decimal. If you need a fractional answer, you might need to recognize common decimal-to-fraction equivalencies (e.g., 0.5 = ¹⁄₂) or use conversion techniques if the decimal is terminating or repeating.
• Scientific or Graphing Calculators: Many scientific and graphing calculators have dedicated fraction buttons (often labeled like a ^b⁄_c or x/y). These allow you to:
  • Enter fractions directly (proper, improper, and sometimes mixed numbers).
  • Perform operations (+, -, ×, ÷) with fractions and get results as fractions (often automatically simplified).
  • Convert between fractions and decimals, and between improper fractions and mixed numbers.
  Consult your calculator's manual for specific instructions.
• Online Fraction Calculators: Numerous websites and apps offer specialized fraction calculators that can perform all these operations and show steps.

The calculator provided on this page is a basic decimal calculator. For fraction operations, you would convert your fractions to decimals first.

"³⁄₄ in half" means dividing ³⁄₄ by 2.

(³⁄₄) ÷ 2 = (³⁄₄) ÷ (²⁄₁) = (³⁄₄) × (¹⁄₂) = ^{(3 × 1)}⁄_{(4 × 2)} = ³⁄₈

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"³⁄₄ of a half" means multiplying ³⁄₄ by ¹⁄₂.

(³⁄₄) × (¹⁄₂) = ^{(3 × 1)}⁄_{(4 × 2)} = ³⁄₈

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"¹⁄₂ of ³⁄₄" also means multiplying ¹⁄₂ by ³⁄₄.

(¹⁄₂) × (³⁄₄) = ^{(1 × 3)}⁄_{(2 × 4)} = ³⁄₈

All three questions result in ³⁄₈.

"Half of ¹⁄₃" means multiplying ¹⁄₂ by ¹⁄₃.

(¹⁄₂) × (¹⁄₃) = ^{(1 × 1)}⁄_{(2 × 3)} = ¹⁄₆

So, half of ¹⁄₃ is ¹⁄₆.

To compare ¹⁄₃ and ¹⁄₂, we can find a common denominator or convert them to decimals.

Method 1: Common Denominator

• The least common multiple of 3 and 2 is 6.
• Convert ¹⁄₃: ¹⁄₃ = ^{(1 × 2)}⁄_{(3 × 2)} = ²⁄₆
• Convert ¹⁄₂: ¹⁄₂ = ^{(1 × 3)}⁄_{(2 × 3)} = ³⁄₆
• Now compare ²⁄₆ and ³⁄₆. Since 2 is less than 3, ²⁄₆ is less than ³⁄₆.

Method 2: Convert to Decimals

• ¹⁄₃ = 1 ÷ 3 = 0.333... (repeating)
• ¹⁄₂ = 1 ÷ 2 = 0.5
• Comparing 0.333... and 0.5, we see that 0.333... is less than 0.5.

Therefore, No, ¹⁄₃ is not more than ¹⁄₂; it is less than ¹⁄₂.

To compare ³⁄₈ and ¹⁄₄, we can find a common denominator or convert them to decimals.

Method 1: Common Denominator

• The least common multiple of 8 and 4 is 8.
• ³⁄₈ already has the denominator 8.
• Convert ¹⁄₄: ¹⁄₄ = ^{(1 × 2)}⁄_{(4 × 2)} = ²⁄₈
• Now compare ³⁄₈ and ²⁄₈. Since 3 is greater than 2, ³⁄₈ is greater than ²⁄₈.

Method 2: Convert to Decimals

• ³⁄₈ = 3 ÷ 8 = 0.375
• ¹⁄₄ = 1 ÷ 4 = 0.25
• Comparing 0.375 and 0.25, we see that 0.375 is greater than 0.25.

Therefore, Yes, ³⁄₈ is more than ¹⁄₄.

These phrases can be ambiguous, but in a mathematical context, especially concerning fractions, here are common interpretations:

"What is 3 2?"

• If meant as a fraction (common in spoken math problems): It likely means ³⁄₂ (three-halves).

  ³⁄₂ = 1.5 = 1 ¹⁄₂

• It could also mean multiplication: 3 × 2 = 6.
• Or simply the numbers 3 and 2.

Given the FAQ list context, ³⁄₂ is the most probable interpretation.

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"What's 2 3 times 2?"

• Most likely interpreted as "What is ²⁄₃ times 2?":

  (²⁄₃) × 2 = (²⁄₃) × (²⁄₁) = ^{(2 × 2)}⁄_{(3 × 1)} = ⁴⁄₃

  ⁴⁄₃ = 1 ¹⁄₃ ≈ 1.333...

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"What is 2 3 2 3?"

This is highly ambiguous. Possible interpretations:

• As multiplication of fractions ²⁄₃ and ²⁄₃:

  (²⁄₃) × (²⁄₃) = ^{(2 × 2)}⁄_{(3 × 3)} = ⁴⁄₉

  ⁴⁄₉ ≈ 0.444...

• As addition of fractions ²⁄₃ + ²⁄₃:

  ²⁄₃ + ²⁄₃ = ⁽²⁺²⁾⁄₃ = ⁴⁄₃ = 1 ¹⁄₃

• As a sequence or pattern.

In lists of math problems, repeated values like this often imply multiplication, so ⁴⁄₉ is a strong candidate if it's a fraction context.

"Half of 1 ¹⁄₂" means multiplying ¹⁄₂ by the mixed number 1 ¹⁄₂.

Steps:

1. Convert the mixed number to an improper fraction:

   1 ¹⁄₂ = ^{( (1 × 2) + 1 )}⁄₂ = ³⁄₂

2. Multiply ¹⁄₂ by the improper fraction:

   (¹⁄₂) × (³⁄₂) = ^{(1 × 3)}⁄_{(2 × 2)} = ³⁄₄

So, half of 1 ¹⁄₂ is ³⁄₄.

The mole fraction (usually denoted by X or χ Greek letter chi) is a way of expressing the concentration of a component in a mixture. It is the ratio of the number of moles of that component to the total number of moles of all components in the mixture.

Formula:

For a component i in a mixture, its mole fraction X_i is:

X_i = ^{Number of moles of component i (n_i)}⁄_{Total number of moles of all components (ntotal)}

Or, more simply:

X_i = n_i ⁄ n_total

Where n_total = n₁ + n₂ + n₃ + ... (the sum of moles of all components).

Key Properties of Mole Fractions:

• Mole fractions are dimensionless (they have no units).
• The sum of the mole fractions of all components in a mixture is always equal to 1.

  Σ X_i = X₁ + X₂ + X₃ + ... = 1

• The mole fraction of a component must be between 0 and 1 (inclusive).

Example:

A mixture contains 2 moles of ethanol (C₂H₅OH) and 8 moles of water (H₂O).

1. Moles of ethanol (n_ethanol) = 2 mol
2. Moles of water (n_water) = 8 mol
3. Total moles (n_total) = n_ethanol + n_water = 2 mol + 8 mol = 10 mol
4. Mole fraction of ethanol (X_ethanol):

   X_ethanol = ^n_ethanol⁄_ntotal = ^{2 mol}⁄_{10 mol} = 0.2

5. Mole fraction of water (X_water):

   X_water = ^n_water⁄_ntotal = ^{8 mol}⁄_{10 mol} = 0.8

Check: X_ethanol + X_water = 0.2 + 0.8 = 1.0.