GCSE Maths Practice: standard-form

Ratios and Division in Standard Form – Computer Science Example

Standard form is widely used in computing and digital technology to describe very large and very small quantities, such as data rates, frequencies, and storage capacities. Comparing these quantities often requires forming ratios. Using standard form allows engineers and analysts to express and compare magnitudes without handling long strings of zeros.

Understanding the Concept

A ratio compares how many times one value contains another. In standard form, division follows the law of indices:

\[ \dfrac{a_1 \times 10^{n_1}}{a_2 \times 10^{n_2}} = \left( \dfrac{a_1}{a_2} \right) \times 10^{n_1 - n_2}. \]

Dividing numbers in standard form therefore involves two steps: dividing the coefficients and subtracting the exponents. The result should always be written so that the coefficient lies between 1 and 10 to maintain correct standard form.

Step-by-Step Strategy

1. Identify both numbers in standard form.
2. Divide the coefficients using normal arithmetic.
3. Subtract the powers of ten according to the index law.
4. Adjust the coefficient if necessary so that \( 1 \leq a < 10 \).
5. Write the final result clearly in standard form.

Worked Example 1: Comparing Transfer Speeds

Suppose an old USB 2.0 port transfers data at \( 4.8 \times 10^8 \text{ bits/s} \), while a modern fibre connection reaches \( 9.6 \times 10^{11} \text{ bits/s} \). To find how many times faster the fibre connection is:

\[ \dfrac{9.6 \times 10^{11}}{4.8 \times 10^8} = (2.0) \times 10^{3} = 2.0 \times 10^3. \]

This shows the fibre link is about 2000 times faster.

Worked Example 2: Memory Capacity Ratio

A computer with a memory of \( 3.2 \times 10^9 \text{ bytes} \) is compared with a smaller device holding \( 8.0 \times 10^7 \text{ bytes} \). The ratio is:

\[ \dfrac{3.2 \times 10^9}{8.0 \times 10^7} = 0.4 \times 10^{2} = 4.0 \times 10^1. \]

The computer has 40 times more memory.

Common Mistakes

• Adding exponents instead of subtracting: Remember, in division the exponents are subtracted.
• Leaving coefficients outside the correct range: Always check that the number before the power of ten is between 1 and 10.
• Ignoring units: Ensure both quantities use the same units (bits, bytes, or hertz) before dividing.

Real-World Relevance

In computing, standard form helps when expressing file sizes, bandwidths, and processor speeds that span multiple magnitudes. For instance, a processor operating at \( 3.6 \times 10^9 \text{ Hz} \) (3.6 GHz) compared to an older 486 processor at \( 6.0 \times 10^6 \text{ Hz} \) has a ratio of:

\[ \dfrac{3.6 \times 10^9}{6.0 \times 10^6} = 0.6 \times 10^{3} = 6.0 \times 10^2. \]

This shows the modern CPU is around 600 times faster.

FAQs

Q1: Why subtract powers when dividing?
A: Because \( 10^a \div 10^b = 10^{a-b} \). The difference between the exponents shows how many factors of ten separate the values.

Q2: Can ratios in computing be less than one?
A: Yes. If the numerator is smaller, the result has a coefficient below 1, which you must adjust back into standard form.

Q3: Does the unit cancel out in ratios?
A: When both quantities share the same unit (e.g., bits per second), the ratio itself is dimensionless—it simply compares magnitudes.

Study Tip

Always perform ratio calculations carefully: divide coefficients separately from exponents. In computer science, this technique is vital for comparing speeds, storage, and signal strengths expressed in scientific notation. Showing clear steps ensures full marks in GCSE Higher exams involving standard form and powers of ten.