GCSE Maths Practice: standard-form

Dividing Numbers in Standard Form – Engineering Example

Standard form is not only used in astronomy and chemistry—it also appears throughout engineering, electronics, and computer science. In this problem, an electrical generator produces \( 8.4 \times 10^7 \) watts of power, and each smaller circuit draws \( 2.1 \times 10^3 \) watts. Dividing these quantities reveals how many circuits the generator can supply simultaneously.

The Core Rule

When dividing numbers written in standard form, use the law of indices:

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

This means you divide the coefficients and subtract the exponents. The result should always be rewritten so that the coefficient \( a \) lies between 1 and 10, ensuring it is in true standard form.

Step-by-Step Method

1. Write both quantities in standard form. Here we already have \( 8.4 \times 10^7 \) and \( 2.1 \times 10^3 \).
2. Divide the coefficients: \( 8.4 \div 2.1 = 4.0 \).
3. Subtract the powers: \( 7 - 3 = 4 \).
4. Combine: \( 4.0 \times 10^4 \).

The generator can therefore supply \( 4.0 \times 10^4 \) circuits, or forty thousand in full notation.

Worked Example 1: Mechanical Power

An engine delivers \( 6.0 \times 10^6 \text{ J} \) of work in \( 1.5 \times 10^2 \text{ s} \). The average power is:

\[ \dfrac{6.0 \times 10^6}{1.5 \times 10^2} = 4.0 \times 10^4 \text{ W}. \]

This is the same operation—division of standard-form quantities.

Worked Example 2: Computer Data Rate

A data server processes \( 2.5 \times 10^{10} \) bits in \( 5.0 \times 10^5 \) seconds. The transfer rate is:

\[ \dfrac{2.5 \times 10^{10}}{5.0 \times 10^5} = 0.5 \times 10^5 = 5.0 \times 10^4 \text{ bits/s}. \]

Common Mistakes

• Adding instead of subtracting exponents: Remember that division means subtracting the powers.
• Leaving the coefficient outside 1–10 range: If your coefficient is greater than 10, move the decimal one place left and increase the exponent by 1.
• Dropping significant figures: Keep the result to the same precision as the data given (usually 2–3 s.f. in GCSE).

Real-World Uses

Engineers use standard form when expressing electrical current (amperes), energy (joules), and charge (coulombs), since these values often span many magnitudes. For instance, microcurrents in sensors might be \( 1.2 \times 10^{-6} \text{ A} \), while large-scale generators produce \( 3.5 \times 10^8 \text{ W} \). Without standard form, comparing such values would be slow and error-prone.

FAQs

Q1: Why subtract powers when dividing?
A: Because \( 10^a \div 10^b = 10^{a-b} \). This rule comes directly from the laws of indices.

Q2: What if the numbers are not in standard form initially?
A: Convert them first. For example, \( 42000 = 4.2 \times 10^4 \).

Q3: How do I check my answer?

A: Multiply your result by the divisor—it should reproduce the original numerator if correct.

Study Tip

Always perform coefficient division and exponent subtraction separately to avoid confusion. Write every step clearly and confirm your coefficient lies between 1 and 10 before finalising. This structured method ensures accuracy on GCSE Higher problems involving division in standard form.