Fractions and fraction calculations

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课程笔记

What is a fraction?

A fraction represents a part of a whole or any number of equal parts.
A simple fraction is written as \frac{a}{b}, where a is the numerator (top number) and b is the denominator (bottom number).
The denominator must not be zero.
The numerator counts how many equal parts are taken; the denominator says how many equal parts make a whole.
For example, \frac{3}{4} means 3 equal parts out of 4 that make a whole.
Fractions can also represent division: \frac{3}{4} means 3 ÷ 4.
Fractions can represent ratios: \frac{3}{4} is the ratio 3:4.

Equivalent fractions and simplifying

Equivalent fractions are different fractions that represent the same value, e.g. \frac{1}{2} = \frac{2}{4} = \frac{3}{6}.
To make an equivalent fraction, multiply or divide both numerator and denominator by the same non-zero number.
Simplifying (or reducing) a fraction means dividing numerator and denominator by their highest common factor (HCF) until they are coprime (no common factor other than 1).
A fraction is in simplest form when the numerator and denominator have no common factors other than 1.
Example: \frac{8}{12} simplifies to \frac{2}{3} by dividing both by 4.

Improper fractions and mixed numbers

An improper fraction has a numerator greater than or equal to its denominator, e.g. \frac{7}{4}.
A mixed number consists of a whole number and a proper fraction, e.g. 1\frac{3}{4}.
To convert an improper fraction to a mixed number: divide the numerator by the denominator; the quotient is the whole number and the remainder over the denominator is the fraction part.
Example: \frac{7}{4} = 1\frac{3}{4} because 7 ÷ 4 = 1 remainder 3.
To convert a mixed number to an improper fraction: multiply the whole number by the denominator, add the numerator, and place over the denominator.
Example: 2\frac{1}{3} = \frac{7}{3} because 2 × 3 + 1 = 7.

Comparing and ordering fractions

To compare fractions with the same denominator, compare the numerators: the larger numerator is the larger fraction.
To compare fractions with different denominators, first write them with a common denominator (often the lowest common multiple of the denominators).
Alternatively, convert each fraction to a decimal and compare the decimals.
Example: to compare \frac{2}{3} and \frac{3}{4}, use denominator 12: \frac{8}{12} and \frac{9}{12}, so \frac{3}{4} is larger.
When ordering, arrange from smallest to largest (or as required) after converting to a common denominator.

Adding and subtracting fractions

To add or subtract fractions with the same denominator, add or subtract the numerators and keep the denominator.
To add or subtract fractions with different denominators, first find a common denominator.
The common denominator is a common multiple of the denominators; the lowest common multiple (LCM) is often used.
Convert each fraction to an equivalent fraction with the common denominator, then add or subtract the numerators.
Simplify the result if possible.
Example: \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}.
For mixed numbers, add or subtract the whole parts and the fraction parts separately, regrouping if needed.

Multiplying fractions

To multiply fractions, multiply the numerators together and multiply the denominators together.
Simplify the result if possible.
Example: \frac{2}{3} \times \frac{4}{5} = \frac{8}{15}.
To multiply mixed numbers, first convert them to improper fractions, then multiply.
Example: 1\frac{1}{2} \times \frac{2}{3} = \frac{3}{2} \times \frac{2}{3} = \frac{6}{6} = 1.
Cancelling common factors before multiplying can make the calculation easier.

Dividing fractions

To divide by a fraction, multiply by its reciprocal (turn the second fraction upside down).
Example: \frac{1}{2} \div \frac{3}{4} = \frac{1}{2} \times \frac{4}{3} = \frac{4}{6} = \frac{2}{3}.
To divide mixed numbers, convert them to improper fractions first, then multiply by the reciprocal.
Example: 2\frac{1}{2} \div \frac{1}{3} = \frac{5}{2} \times \frac{3}{1} = \frac{15}{2} = 7\frac{1}{2}.
Simplify the result if possible.

Finding a fraction of an amount

To find a fraction of an amount, multiply the amount by the fraction.
Example: to find \frac{3}{4} of 20, calculate 20 \times \frac{3}{4} = 15.
Alternatively, divide the amount by the denominator, then multiply by the numerator.
Example: \frac{3}{4} of 20: 20 ÷ 4 = 5, then 5 × 3 = 15.
This method works for any fraction of a quantity.

Fractions in word problems

Identify the whole amount and the fraction being used.
For 'fraction of' problems, multiply the whole amount by the fraction.
For addition or subtraction problems, ensure the fractions have a common denominator before combining.
For multiplication or division problems, convert mixed numbers to improper fractions first.
Always check that the answer makes sense in the context of the problem.
Simplify fractions in your final answer where possible.

幻灯片

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练习题

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1.In the fraction \frac{3}{4}, which number is the denominator?
Easy
- A3
- B4
- C7
- D12
2.What name is given to the top number of a fraction, which counts how many equal parts are being described?
Easy
- ADenominator
- BNumerator
- CQuotient
- DDivisor
3.The fraction \frac{3}{4} can also be written as a division. Which calculation is correct?
Easy
- A4 \div 3
- B3 \div 4
- C3 \times 4
- D4 - 3
4.Which of these fractions is equivalent to \frac{2}{3}?
Medium
- A\frac{3}{4}
- B\frac{4}{6}
- C\frac{6}{8}
- D\frac{2}{6}
5.Which of these fractions is equivalent to \frac{6}{8}?
Medium
- A\frac{2}{3}
- B\frac{3}{4}
- C\frac{4}{6}
- D\frac{1}{2}
6.Which of the following fractions is the largest?
Medium
- A\frac{2}{3}
- B\frac{3}{5}
- C\frac{7}{10}
- D\frac{1}{2}
7.Which of the following fractions is the smallest?
Medium
- A\frac{3}{5}
- B\frac{2}{3}
- C\frac{5}{8}
- D\frac{1}{2}
8.Work out \frac{1}{4} + \frac{1}{3}.
Medium
- A\frac{2}{7}
- B\frac{7}{12}
- C\frac{7}{24}
- D\frac{1}{12}

复习笔记

## Fractions and Fraction Calculations A **fraction** represents a part of a whole or any number of equal parts. In a simple fraction like \(\frac{3}{4}\), the **numerator** (top) tells you how many equal parts, and the **denominator** (bottom) tells you how many parts make a whole. The denominator is never zero. Fractions also represent **division** and **ratio**: \(\frac{3}{4}\) means 3 ÷ 4 and the ratio 3:4. ### Equivalent fractions and simplifying Multiply or divide the numerator and denominator by the same non-zero number to get an **equivalent fraction**. To simplify, divide both by their highest common factor. ### Improper fractions and mixed numbers An **improper fraction** has a numerator greater than or equal to its denominator (e.g. \(\frac{17}{3}\)). A **mixed number** is a whole number and a fraction (e.g. \(5\frac{2}{3}\)). Convert: \(\frac{17}{3} = 5\frac{2}{3}\) because 17 ÷ 3 = 5 remainder 2. ### Comparing and ordering To compare fractions, rewrite them with a common denominator, then compare numerators. ### Adding and subtracting Rewrite with a common denominator, then add or subtract the numerators. For mixed numbers, convert to improper fractions first. ### Multiplying and dividing - Multiply: multiply numerators together and denominators together. Simplify if possible. - Divide: multiply by the reciprocal (flip the second fraction). ### Fraction of an amount To find a fraction of an amount, divide by the denominator and multiply by the numerator. For example, \(\frac{3}{4}\) of 20 = 20 ÷ 4 × 3 = 15. ### Negative fractions A negative fraction is the opposite of a positive one. \(-\frac{1}{2}\), \(\frac{-1}{2}\) and \(\frac{1}{-2}\) are all the same: negative one-half. \(\frac{-1}{-2}\) is positive one-half. ### Rational numbers A **rational number** can be written as \(\frac{a}{b}\), where a and b are integers and b ≠ 0. The set of rational numbers is denoted by Q (for quotient). ### Word problems Identify the fraction and the whole amount. Decide whether you need a fraction of an amount, addition, subtraction, multiplication or division, then apply the rules above.

Fraction: A number that represents a part of a whole or any number of equal parts, written as a numerator over a denominator.
Numerator: The top number of a fraction; it counts how many equal parts are being described.
Denominator: The bottom number of a fraction; it tells how many equal parts make up a whole. It cannot be zero.
Equivalent fractions: Fractions that represent the same value, obtained by multiplying or dividing numerator and denominator by the same non-zero number.
Improper fraction: A fraction whose numerator is greater than or equal to its denominator, e.g. \(\frac{17}{3}\).
Mixed number: A number written as a whole number and a proper fraction, e.g. \(5\frac{2}{3}\).
Reciprocal: The fraction obtained by swapping numerator and denominator; used when dividing fractions.
Rational number: A number that can be written as \(\frac{a}{b}\), where a and b are integers and b ≠ 0. The set is denoted Q.

闪卡

What does the numerator of a fraction represent?

The numerator is the number of equal parts being described. In 3/4, the numerator 3 means 3 equal parts.

What does the denominator of a fraction represent?

The denominator is the type or size of the parts, i.e. how many equal parts make up a whole. In 3/4, the denominator 4 means 4 parts make up a whole.

What is an equivalent fraction?

Equivalent fractions are different fractions that represent the same value, e.g. 1/2 = 2/4 = 3/6. They are made by multiplying or dividing the numerator and denominator by the same non-zero number.

How do you simplify a fraction?

Divide the numerator and denominator by their highest common factor. For example, 12/18 simplifies to 2/3 because 12 and 18 have HCF 6.

What is an improper fraction?

An improper fraction has a numerator greater than or equal to its denominator, e.g. 7/4. It represents a value of 1 or more.

What is a mixed number?

A mixed number is a whole number and a proper fraction written together, e.g. 1 3/4. It is equivalent to an improper fraction.

Convert the improper fraction 17/5 to a mixed number.

17 ÷ 5 = 3 remainder 2, so 17/5 = 3 2/5.

Convert the mixed number 2 3/7 to an improper fraction.

2 3/7 = (2 × 7 + 3)/7 = 17/7.

How do you compare two fractions with different denominators?

Write them with a common denominator, then compare numerators. For example, to compare 2/3 and 3/4, use 12: 8/12 and 9/12, so 3/4 is larger.

How do you add fractions with different denominators?

Find a common denominator, rewrite each fraction as an equivalent fraction, then add the numerators. For example, 1/2 + 1/3 = 3/6 + 2/6 = 5/6.

How do you subtract fractions with different denominators?

Find a common denominator, rewrite each fraction, then subtract the numerators. For example, 3/4 − 1/3 = 9/12 − 4/12 = 5/12.

How do you multiply two fractions?

Multiply the numerators together and multiply the denominators together. For example, 2/3 × 4/5 = 8/15.

How do you divide by a fraction?

Multiply by its reciprocal (turn the second fraction upside down). For example, 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8.

How do you multiply mixed numbers?

Change each mixed number to an improper fraction first, then multiply. For example, 1 1/2 × 2 1/3 = 3/2 × 7/3 = 21/6 = 3 1/2.

How do you divide mixed numbers?

Change both to improper fractions, then multiply by the reciprocal of the second. For example, 2 1/2 ÷ 1 1/4 = 5/2 ÷ 5/4 = 5/2 × 4/5 = 2.

How do you find a fraction of an amount?

Divide the amount by the denominator, then multiply by the numerator. For example, 3/4 of 20 = 20 ÷ 4 × 3 = 15.

What does the fraction bar also represent?

The fraction bar represents division. For example, 3/4 means 3 ÷ 4.

How can a fraction represent a ratio?

The fraction 3/4 can represent the ratio 3:4, comparing the part to the whole.

What is a rational number?

A rational number is any number that can be written as a fraction a/b, where a and b are integers and b is not zero. The set of rational numbers is denoted by Q.

What is the reciprocal of a fraction?

The reciprocal of a/b is b/a. For example, the reciprocal of 3/5 is 5/3.

How do you add mixed numbers?

Add the whole numbers and add the fractions separately, then combine. For example, 1 1/4 + 2 1/2 = 3 + 3/4 = 3 3/4.

How do you subtract mixed numbers?

Subtract the whole numbers and subtract the fractions, borrowing if needed. For example, 3 1/2 − 1 3/4 = 7/2 − 7/4 = 14/4 − 7/4 = 7/4 = 1 3/4.

What is a proper fraction?

A proper fraction has a numerator smaller than its denominator, e.g. 3/5. Its value is less than 1.

How do you order fractions with the same denominator?

Compare the numerators: the larger the numerator, the larger the fraction. For example, 5/7 > 3/7.

How do you order fractions with the same numerator?

Compare the denominators: the smaller the denominator, the larger the fraction. For example, 3/4 > 3/5.

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