Fractions, decimals and percentages Study Guide

Equivalent fractions are different ways of writing the same value. They represent the same proportion of a whole, even though they might have different numerators (the top number) and denominators (the bottom number). For example, a pizza cut into 2 equal slices, with 1 slice eaten, represents half the pizza. If the same pizza is cut into 4 equal slices, and 2 slices are eaten, it still represents half the pizza. So, 1/2 and 2/4 are equivalent fractions.

To find an equivalent fraction, you multiply or divide both the numerator and the denominator by the same non-zero number. This operation is valid because it's essentially multiplying the fraction by 1 (e.g., 2/2 = 1, 3/3 = 1), which does not change the fraction's value. When you multiply, you are essentially cutting each existing 'piece' into smaller, equal 'sub-pieces', increasing both the total number of pieces and the number of pieces you have by the same factor. When you divide, you are grouping existing 'pieces' into larger 'super-pieces', decreasing both the total number of pieces and the number of pieces you have by the same factor. The key is that the *proportion* remains constant.

For instance, to find a fraction equivalent to 3/4 with a denominator of 12:

1. Observe that the denominator 4 was multiplied by 3 to get 12 (since 4 x 3 = 12).
2. To keep the fraction equivalent, the numerator must also be multiplied by 3.
3. So, 3 x 3 = 9.
4. Therefore, 9/12 is equivalent to 3/4.

Similarly, to simplify a fraction like 10/15:

1. Find a common factor for both 10 and 15. Both are divisible by 5.
2. Divide both the numerator and the denominator by 5.
3. 10 ÷ 5 = 2 and 15 ÷ 5 = 3.
4. Therefore, 2/3 is equivalent to 10/15 and is in its simplest form.

Fractions represent parts of a whole. A fraction consists of a numerator (the top number) and a denominator (the bottom number). The denominator tells us how many equal parts the whole is divided into, and the numerator tells us how many of those parts we are considering. For example, in the fraction 3/4, the whole is divided into 4 equal parts, and we are looking at 3 of those parts.

Before performing operations like addition or subtraction, fractions often need a 'common denominator'. This means rewriting the fractions so they both have the same bottom number. To do this, you find the least common multiple (LCM) of the denominators and then multiply both the numerator and denominator of each fraction by the factor that makes the denominator equal to the LCM. For example, to add 1/2 and 1/3, the LCM of 2 and 3 is 6. So, 1/2 becomes 3/6 (multiplying top and bottom by 3), and 1/3 becomes 2/6 (multiplying top and bottom by 2). Then, you can add 3/6 + 2/6 = 5/6.

Multiplication of fractions is simpler: multiply the numerators together and multiply the denominators together. For example, (1/2) * (3/4) = (1*3)/(2*4) = 3/8.

Division of fractions involves a trick: 'keep, change, flip'. Keep the first fraction as it is, change the division sign to multiplication, and flip the second fraction (find its reciprocal by swapping numerator and denominator). Then proceed with multiplication. For example, (1/2) / (3/4) = (1/2) * (4/3) = (1*4)/(2*3) = 4/6, which simplifies to 2/3.

Understanding these fundamental methods is key to tackling more complex problems involving fractions.