GCSE Maths Practice: standard-form

Understanding Ratios in Standard Form

Dividing or comparing very large or small numbers is a core part of GCSE Maths, particularly within the topic of Standard Form. When we compare astronomical or microscopic quantities, the values are often written as powers of ten. Learning to divide such numbers efficiently makes calculations simpler and highlights the true scale of the quantities being compared.

The Concept

Any number written in standard form has the structure \( a \times 10^n \), where \( 1 \leq a < 10 \) and \( n \) is an integer. To divide two numbers in standard form, divide their coefficients and subtract the exponents. This follows directly from the laws of indices:

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

If the resulting coefficient is less than 1 or greater than or equal to 10, you must adjust it to return to proper standard form by moving the decimal point and modifying the power of ten accordingly.

Step-by-Step Method

1. Write both numbers in standard form, clearly identifying the coefficient and power of ten.
2. Divide the coefficients using normal arithmetic.
3. Subtract the exponents: \( n_1 - n_2 \).
4. Adjust the result so the coefficient lies between 1 and 10.
5. Express the final answer neatly in standard form.

Worked Example 1

Compare \( 9.6 \times 10^7 \) and \( 3.2 \times 10^4 \).
\[ \dfrac{9.6 \times 10^7}{3.2 \times 10^4} = 3 \times 10^{7-4} = 3 \times 10^3. \]
This means the first value is three thousand times larger.

Worked Example 2

Compare \( 2.4 \times 10^{-6} \) and \( 8.0 \times 10^{-9} \).
\[ \dfrac{2.4 \times 10^{-6}}{8.0 \times 10^{-9}} = 0.3 \times 10^{3} = 3.0 \times 10^2. \]
The first quantity is 300 times greater.

Applying to Astronomy

In this question, we compare the distance from Earth to Mars (\(2.25 \times 10^8\) km) with the distance from Earth to the Moon (\(3.84 \times 10^5\) km). The calculation shows that Mars is roughly \(5.86 \times 10^2\) times farther away than the Moon. That is about 586 times farther — a powerful example of how standard form quickly conveys immense differences in scale.

Common Errors

• Adding exponents instead of subtracting: Remember, division means subtracting powers of ten.
• Leaving the coefficient outside the 1–10 range: Always adjust the decimal and exponent to restore standard form.
• Forgetting to maintain units: Ratios themselves are unitless, but ensure the two quantities being compared share the same unit before dividing.

Real-World Connections

Standard form division appears in physics (calculating ratios of force, distance, or luminosity), in chemistry (comparing concentrations), and even in economics when analysing quantities that differ by millions or billions. The process allows scientists and analysts to compare values instantly, regardless of scale.

FAQs

Q1: Why do we subtract powers when dividing?
A: Because \(10^a \div 10^b = 10^{a-b}\) according to the index laws.

Q2: What if the numbers have negative powers?
A: The same rule applies — subtract the exponents carefully and keep track of signs.

Q3: Can the ratio ever be less than one?
A: Yes. If the numerator is smaller than the denominator, the coefficient will be less than 1 and the power may become negative after adjustment.

Study Tip

Always align both numbers in standard form before dividing. Perform the coefficient division separately from the power operation. Write all steps clearly — GCSE exam questions often award method marks for showing how you subtracted exponents or adjusted the final form.