Fractions, decimals and percentages Revision Notes

What are Equivalent Fractions?

Equivalent fractions are different representations of the same fractional value. They convey the same proportion or part of a whole, even if their numerator and denominator are different numbers. Think of it as having the same amount of a pie, but the pie might be cut into a different number of slices. For example, 1/2, 2/4, 3/6, and 5/10 all represent the same amount – half of something. The key idea is that the ratio between the numerator and the denominator remains constant. Understanding equivalent fractions is fundamental for comparing and ordering fractions, as well as for performing arithmetic operations such as addition and subtraction of fractions.

Finding Equivalent Fractions by Multiplication

To find an equivalent fraction, you can multiply both the numerator and the denominator by the exact same non-zero number. Multiplying both parts of a fraction by the same number is like multiplying the fraction by 1 (e.g., 2/2 or 5/5), which doesn't change its value. For example, to find an equivalent fraction for 2/3:

• Multiply both numerator and denominator by 2: (2 × 2) / (3 × 2) = 4/6. So, 2/3 is equivalent to 4/6.
• Multiply both numerator and denominator by 5: (2 × 5) / (3 × 5) = 10/15. So, 2/3 is equivalent to 10/15.

This method allows you to create an unlimited number of equivalent fractions for any given fraction.

Simplifying Fractions by Division (Reducing to Simplest Form)

To simplify a fraction, also known as reducing it to its simplest form, you divide both the numerator and the denominator by their greatest common factor (GCF). The GCF is the largest number that divides exactly into both numbers. When a fraction is in its simplest form, the numerator and denominator have no common factors other than 1. This is often the preferred way to write a fraction. For example, to simplify the fraction 12/18:

1. Find the common factors of 12 and 18: They are 1, 2, 3, and 6.
2. The greatest common factor (GCF) is 6.
3. Divide both the numerator and the denominator by 6: (12 ÷ 6) / (18 ÷ 6) = 2/3.

So, 12/18 simplifies to 2/3. This method is useful for making fractions easier to understand and work with.

Identifying Equivalent Fractions

You can identify if two fractions are equivalent in several ways:

1. Cross-multiplication: Multiply the numerator of the first fraction by the denominator of the second, and the numerator of the second by the denominator of the first. If the products are equal, the fractions are equivalent. For example, for 2/3 and 4/6: (2 × 6) = 12 and (3 × 4) = 12. Since 12 = 12, they are equivalent.
2. Simplifying to simplest form: Reduce both fractions to their simplest form. If their simplest forms are the same, they are equivalent. For example, 10/15 simplifies to 2/3 (divide by 5), and 4/6 simplifies to 2/3 (divide by 2). Since both simplify to 2/3, they are equivalent.
3. Finding a common multiplier/divisor: Check if you can multiply or divide the numerator and denominator of one fraction by a single number to get the other fraction. If yes, they are equivalent.

Understanding Fraction Notation

A fraction is written as a/b, where 'a' is the numerator and 'b' is the denominator. The numerator 'a' tells us how many parts we have, and the denominator 'b' tells us how many parts the whole is divided into. For example, in 5/8, you have 5 parts out of a total of 8 equal parts. A proper fraction has a numerator smaller than its denominator (e.g., 2/3), an improper fraction has a numerator greater than or equal to its denominator (e.g., 7/4), and a mixed number combines a whole number and a proper fraction (e.g., 1 3/4).

Adding and Subtracting Fractions

To add or subtract fractions, they must have a common denominator. Find the least common multiple (LCM) of the denominators to determine the lowest common denominator (LCD). Convert each fraction to an equivalent fraction with the LCD. For example, to add 1/3 and 1/4: the LCM of 3 and 4 is 12. So, 1/3 becomes 4/12 (multiply top and bottom by 4), and 1/4 becomes 3/12 (multiply top and bottom by 3). Then, add the numerators: 4/12 + 3/12 = 7/12. Always simplify your answer to its lowest terms if possible.

Multiplying and Dividing Fractions

To multiply fractions, simply multiply the numerators together and the denominators together. For example, (2/3) * (4/5) = (2*4)/(3*5) = 8/15. You can simplify before multiplying by cancelling common factors diagonally or vertically, which can make the numbers smaller and easier to work with. To divide fractions, use the 'keep, change, flip' method. Keep the first fraction, change the division sign to multiplication, and flip (find the reciprocal of) the second fraction. For example, (1/2) / (1/4) becomes (1/2) * (4/1) = (1*4)/(2*1) = 4/2 = 2. Always simplify your final answer.

Converting Between Mixed Numbers and Improper Fractions

To convert a mixed number to an improper fraction: multiply the whole number by the denominator, then add the numerator. Keep the original denominator. For example, 2 1/3 = (2 * 3 + 1)/3 = 7/3. To convert an improper fraction to a mixed number: divide the numerator by the denominator. The quotient is the whole number, and the remainder is the new numerator over the original denominator. For example, 7/3 = 2 with a remainder of 1, so it is 2 1/3. It is often easier to work with improper fractions in calculations and convert to mixed numbers at the end if required.

Adding and Subtracting Fractions

When adding or subtracting fractions, they must have the same denominator. If they don't, find the least common multiple (LCM) of the denominators to use as the common denominator. Convert each fraction to an equivalent fraction with this new denominator. Then, add or subtract the numerators while keeping the denominator the same. For example, to calculate 2/5 + 1/3, the LCM of 5 and 3 is 15. So, 2/5 becomes 6/15 and 1/3 becomes 5/15. Then, 6/15 + 5/15 = 11/15. Always simplify your answer to its lowest terms if possible.

Multiplying Fractions

Multiplying fractions is simpler than adding or subtracting. You just multiply the numerators together and multiply the denominators together. The result is a new fraction composed of the product of the numerators over the product of the denominators. For example, to calculate (3/4) * (2/5): (3 * 2) / (4 * 5) = 6/20. This fraction can then be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 2, to get 3/10. Always simplify your final answer.

Dividing Fractions

To divide by a fraction, you multiply by its reciprocal. The reciprocal of a fraction is found by flipping the numerator and the denominator. For example, the reciprocal of 3/4 is 4/3. So, to calculate (1/2) / (3/4), first find the reciprocal of 3/4, which is 4/3. Then multiply (1/2) * (4/3) = 4/6. This can be simplified to 2/3. Remember the 'Keep, Change, Flip' (KCF) mnemonic: Keep the first fraction, Change division to multiplication, Flip the second fraction.

Working with Mixed Numbers

Mixed numbers combine a whole number and a fraction (e.g., 2 3/5). To perform calculations, especially multiplication or division, it's often easiest to convert mixed numbers into improper fractions. To do this, multiply the whole number by the denominator and add the numerator; this new value becomes the numerator over the original denominator. For 2 3/5, this is (2 * 5 + 3) / 5 = 13/5. After performing the calculation, if your answer is an improper fraction, you may need to convert it back to a mixed number by dividing the numerator by the denominator to find the whole number, with any remainder forming the new numerator over the original denominator.

Core method

This is Higher-tier content, so solutions should communicate multi-step reasoning precisely. Higher fraction problems combine several operations, mixed numbers and algebraic reasoning. Convert mixed numbers to improper fractions, use a common denominator for addition or subtraction, and multiply by a reciprocal when dividing. Preserve exact values until the final step and cancel common factors to keep arithmetic manageable. The essential approach is to identify the mathematical structure before calculating, retain exact values where possible, and show enough working for each conclusion to be checked independently.

Worked reasoning and checks

Apply the order of operations. For example, 3/4 − (2/3 × 9/10) = 3/4 − 3/5 = 15/20 − 12/20 = 3/20. In a reverse problem, if 5/8 of a quantity is 35, multiply 35 by 8/5 to obtain 56. A negative sign applies to the whole fraction unless brackets show otherwise. Check the final result against the original information, including signs, units, scale and whether the requested form or degree of accuracy has been supplied.