Standard Form

To calculate a number’s overall value, we add or subtract fractions by ensuring they have common denominators. For example, donald drinks 1½ liters of water in the morning and 1½ liters of water in the evening. To calculate how much water donald drinks in a day, we use addition and subtraction of fractions.

In a fraction, the top number indicates how many parts are considered. The bottom number shows how many equal pieces the whole is divided into. For example, in the fraction 1/2, the numerator shows one part, while the denominator represents that the whole is divided into two equal parts.

The process of adding and subtracting fractions involves joining and removing parts of a whole, as defined by fractions. To perform these operations effectively, some rules should be followed.

Rules for Adding Fractions 

• Confirm the denominators are equal: Check the bottom numbers in two fractions are the same. For example, 5⁄10 + 4⁄10 = 9⁄10. Here, the denominators are the same, 10. So, the calculation will be like this:

            5⁄10 + 4⁄10 = 5 + \(\frac{4}{10}\) =  9⁄10.

• If denominators differ, find the least common denominator (LCD) to proceed with addition: To find the least common denominator, we have to list the multiples of the given denominators. For instance, 1⁄4 + 1⁄6. Here, both the denominators are different. So we have to list the multiples of 4 and 6. 

           The multiples of 4 include 4, 8, 12, 16, 20, 24, …

           The multiples of 6 include 6, 12, 18, 24, 30, …

           Among the lists of multiples, 12 is the least common multiple. So 12 is the denominator.

           Now, we need to match the LCD with the fractions. 

           1⁄4 = 1 × \(\frac{3}{4}\) × 3 = 3⁄12

           1⁄6 = 1 × \(\frac{2}{6}\) × 2 = 2⁄12. Now we can add both fractions easily.
 
           3⁄12 + 2⁄12 = 5⁄12.

• Combine mixed numbers: Convert mixed numbers into improper fractions. After that, follow the addition method. For instance,

 
           1 1⁄3 +2 2⁄3 

           1 1⁄3  = 4⁄3

           2 2⁄3  = 8⁄3

           It becomes 4⁄3 + 8⁄3 = 12⁄3 = 4

Rules for Subtracting 

• Subtract the numerators: If the given fractions have the same denominators, then subtract the numerators. For instance, the given fractions are 8⁄3 and 4⁄3.

           8⁄3 – 4⁄3 = 4⁄3

• Use the biggest numerators to subtract fractions: A negative result occurs if you subtract a larger fraction from a smaller one. For example,

            4⁄3 – 8⁄3 = -4⁄3 

            Here, 8⁄3 is the larger fraction.

• Subtract the mixed numbers in a fraction: Change the mixed numbers into improper fractions, and then follow the remaining steps. 

            First, multiply the whole number with the denominator.

            Then add the numerator, we will get the improper fraction. 

             For instance, 

             3 1⁄2 – 1 2⁄3 

             \((3 × 2) + 1\) =\(\frac{7}{2}\) and 

            \( (1 × 3) + 2\) = \(\frac{5}{3}\)

            Subtract these two fractions:

             \(\frac{7}{2}\) – \(\frac{5}{3}\) 

             Next, we have to find the least common denominator of 2 and 3. The result is 6.

             Now we have to match the LCD (the least common denominator) with the fractions. It becomes:

              21⁄6 – 10⁄6 = 11⁄6 = 1 5⁄6.

• Addition of Fractions: The addition of fractions is a simple operation in mathematics. It teaches us to sum the fractions that have the same or different denominators. If the denominators are the same, we only need to add the numerators. If the denominators are different, we have to find the least common denominator. 

           Take a look at this:

   
            We can add 1⁄4 and 2⁄4:

             1⁄4 + 2⁄4 = 3⁄4

             Here, the denominators are the same. Next, we can move on to the next set of fractions.

              3⁄4 + 4⁄4 = 7⁄4.

This sequence illustrates how fractions with the same denominator increase progressively, starting from 1⁄4, 2⁄4, 3⁄4, 4⁄4, and so on. 

• Subtraction of Fractions: Subtraction of fractions involves the subtraction of two fractional values. With this, we can find the difference between two fractions. The common denominators will remain unchanged, and the numerators will be subtracted. 

            For example, 

            5⁄8 – 3⁄8 = 2⁄8

             We can simplify 2⁄8 as 1⁄4. Take a look at this too:

              3⁄4 – 2⁄4 = 1⁄4 or 

              3⁄4 – 1⁄4 = 2⁄4

              \(\frac{2}{4}\) = \(\frac{1}{2}\) 

Performing addition and subtraction of fractions is sometimes tricky. Here are some of the tips and tricks to calculate the fractions using addition or subtraction:

• Always make sure the denominators are the same before adding or subtracting fractions.
   
• Multiply the numerator and denominator by the same number to make denominators equal.
   
• Once denominators match, add or subtract the numerators while keeping the denominator unchanged.
   
• Reduce the resulting fraction to its simplest form by dividing both numerator and denominator by their greatest common factor.
   
• If the numerator is larger than the denominator, convert it into a mixed number for clarity.

Standard form helps simplify very large or very small numbers, making them easier to read and use. It is widely applied in science, technology, finance, and everyday measurements.

Astronomy: Standard form is used to express extremely large distances in space, such as the distance between planets or stars. For example, The distance from Earth to the Sun is approximately 1.5 × 10⁸ km.

Science and chemistry: Very small measurements, like the size of atoms or molecules, are written in standard form for simplicity. For example, The diameter of a hydrogen atom is about 1 × 10⁻¹⁰ m.

Engineering and technology: Engineers use standard form to handle very large numbers, like power output or electrical charge, efficiently in calculations. For example, A power plant generates 3.6 × 10⁶ watts.

Finance and economics: Standard form is useful when dealing with huge sums of money, national budgets, or population statistics. For example, A country’s population may be written as 1.4 × 10⁹ people.

Data storage and computing: Large amounts of digital data, like memory or storage capacity, are expressed in standard form to make numbers manageable. For example, A data center may store 2.5 × 10¹² bytes of information.