Fractions Trainer

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A fraction represents a part of a whole or a division of one number by another. It is written as two numbers separated by a horizontal line (or a slash), where the top number is called the numerator (the number of parts you have) and the bottom number is called the denominator (the total number of equal parts the whole is divided into). For example, in the fraction 1/2, 1 is the numerator and 2 is the denominator.

Simplifying or reducing a fraction means writing it in its simplest form, where the numerator and denominator have no common factors other than 1. To simplify a fraction:

1. Find the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that divides evenly into both numbers.
2. Divide both the numerator and the denominator by their GCD.

The resulting fraction is the simplified form. For example, to simplify 4/8, the GCD of 4 and 8 is 4. Divide both by 4: 4 ÷ 4 = 1 and 8 ÷ 4 = 2. The simplified fraction is 1/2.

Multiplying fractions is straightforward. To multiply two fractions:

1. Multiply the numerators together to get the new numerator.
2. Multiply the denominators together to get the new denominator.
3. Simplify the resulting fraction if possible.

Example: (1/2) * (3/4) = (1*3) / (2*4) = 3/8.
To multiply a fraction by a whole number, write the whole number as a fraction with a denominator of 1 (e.g., 5 is 5/1), then multiply as usual. Example: (2/3) * 5 = (2/3) * (5/1) = (2*5) / (3*1) = 10/3.

Dividing fractions involves a simple trick: "Keep, Change, Flip".

1. Keep the first fraction as it is.
2. Change the division sign to a multiplication sign.
3. Flip (find the reciprocal of) the second fraction (swap its numerator and denominator).
4. Now, multiply the two fractions as you normally would.
5. Simplify the result if possible.

Example: (1/2) ÷ (3/4) = Keep 1/2, Change ÷ to *, Flip 3/4 to 4/3. Now multiply: (1/2) * (4/3) = (1*4) / (2*3) = 4/6. Simplify 4/6 to 2/3.
To divide a fraction by a whole number, write the whole number as a fraction (e.g., 3 is 3/1), then use Keep, Change, Flip. Example: (2/5) ÷ 3 = (2/5) ÷ (3/1) = (2/5) * (1/3) = 2/15.
To divide a whole number by a fraction, write the whole number as a fraction, then use Keep, Change, Flip. Example: 4 ÷ (1/3) = (4/1) ÷ (1/3) = (4/1) * (3/1) = 12/1 = 12.

Adding fractions requires them to have a common denominator (the same bottom number).

If the denominators are already the same:

1. Add the numerators together.
2. Keep the denominator the same.
3. Simplify the result if possible.

Example: (1/4) + (2/4) = (1+2)/4 = 3/4.

If the denominators are different:

1. Find the least common multiple (LCM) of the denominators. This will be your least common denominator (LCD).
2. Convert each fraction into an equivalent fraction with the LCD as the new denominator. To do this, multiply both the numerator and denominator of each fraction by the factor needed to change its original denominator into the LCD.
3. Once the fractions have the same denominator, add the numerators as described above.
4. Keep the denominator the same.
5. Simplify the result if possible.

Example: (1/3) + (1/6). The LCM of 3 and 6 is 6. Convert 1/3 to have a denominator of 6: Multiply numerator and denominator by 2 (since 3 * 2 = 6). (1*2)/(3*2) = 2/6. Now add: (2/6) + (1/6) = (2+1)/6 = 3/6. Simplify 3/6 to 1/2.

Subtracting fractions also requires a common denominator, similar to addition.

If the denominators are already the same:

1. Subtract the second numerator from the first numerator.
2. Keep the denominator the same.
3. Simplify the result if possible.

Example: (3/5) - (1/5) = (3-1)/5 = 2/5.

If the denominators are different:

1. Find the least common denominator (LCD) of the denominators.
2. Convert each fraction into an equivalent fraction with the LCD.
3. Once the fractions have the same denominator, subtract the numerators.
4. Keep the denominator the same.
5. Simplify the result if possible.

Example: (1/2) - (1/4). The LCD of 2 and 4 is 4. Convert 1/2 to have a denominator of 4: Multiply numerator and denominator by 2. (1*2)/(2*2) = 2/4. Now subtract: (2/4) - (1/4) = (2-1)/4 = 1/4.

To subtract fractions with whole numbers or mixed numbers, it's often easiest to convert everything to improper fractions first, find a common denominator, subtract, and then convert back to a mixed number if needed.

To convert a fraction to a decimal, divide the numerator by the denominator.

Example: To convert 3/4 to a decimal, divide 3 by 4. 3 ÷ 4 = 0.75.
Example: To convert 1/3 to a decimal, divide 1 by 3. 1 ÷ 3 = 0.333... (a repeating decimal).

To convert a decimal to a fraction:

1. Look at the number of decimal places. The number of decimal places tells you the denominator: one place is 10, two places is 100, three places is 1000, and so on (1 followed by that many zeros).
2. Write the decimal number (without the decimal point) as the numerator.
3. Use the corresponding power of 10 (10, 100, 1000, etc.) as the denominator.
4. Simplify the resulting fraction if possible.

Example: Convert 0.75 to a fraction. There are two decimal places, so the denominator is 100. The numerator is 75. The fraction is 75/100. Simplify by dividing both by their GCD, which is 25: 75 ÷ 25 = 3, 100 ÷ 25 = 4. The fraction is 3/4.
Example: Convert 0.125 to a fraction. Three decimal places, so denominator is 1000. Numerator is 125. Fraction is 125/1000. Simplify by dividing both by their GCD, which is 125: 125 ÷ 125 = 1, 1000 ÷ 125 = 8. The fraction is 1/8.

An improper fraction is a fraction where the numerator is greater than or equal to the denominator. For example, 5/4, 7/3, and 2/2 are improper fractions. These represent a value equal to or greater than 1.

A mixed number combines a whole number and a fraction (e.g., 1 1/4). To convert an improper fraction to a mixed number:

1. Divide the numerator by the denominator. The whole number part of the result is the whole number part of the mixed number.
2. The remainder of the division becomes the numerator of the fraction part.
3. The original denominator remains the denominator of the fraction part.
4. Simplify the fraction part if necessary.

Example: Convert 7/3 to a mixed number. Divide 7 by 3. 7 ÷ 3 = 2 with a remainder of 1. The whole number is 2, the new numerator is 1, and the denominator is 3. So, 7/3 is equal to the mixed number 2 1/3.

To convert a mixed number to an improper fraction:

1. Multiply the whole number part by the denominator of the fraction part.
2. Add the numerator of the fraction part to the result from step 1. This becomes the new numerator of the improper fraction.
3. Keep the original denominator as the denominator of the improper fraction.

Example: Convert the mixed number 2 1/3 to an improper fraction. Multiply the whole number (2) by the denominator (3): 2 * 3 = 6. Add the numerator (1) to this result: 6 + 1 = 7. This is the new numerator. Keep the original denominator (3). The improper fraction is 7/3.

Equivalent fractions are fractions that have different numerators and denominators but represent the same value. For example, 1/2, 2/4, 3/6, and 4/8 are all equivalent fractions because they all simplify to 1/2. You can create equivalent fractions by multiplying or dividing both the numerator and denominator by the same non-zero number.

A unit fraction is a fraction where the numerator is 1. It represents one part of a whole that has been divided into equal parts. Examples include 1/2, 1/3, 1/4, 1/100, etc.

To convert a fraction to a percent, first convert the fraction to a decimal by dividing the numerator by the denominator. Then, multiply the decimal by 100 and add the percent symbol (%).

Example: Convert 3/4 to a percent. First, convert 3/4 to a decimal: 3 ÷ 4 = 0.75. Then, multiply the decimal by 100: 0.75 * 100 = 75. Add the percent symbol: 75%. So, 3/4 is equal to 75%.