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Number - Ordering

Grade 9IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Ordering numbers involves arranging them in ascending order (smallest to largest) or descending order (largest to smallest) using inequality symbols.

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The inequality symbols are: << (less than), >> (greater than), ≤\leq (less than or equal to), and ≥\geq (greater than or equal to).

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To compare different formats (fractions, decimals, percentages), convert all numbers into a single format, usually decimals.

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For negative numbers, the number with the larger absolute magnitude is smaller. For example, −10<−5-10 < -5.

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When comparing numbers in standard form A×10nA \times 10^n, the number with the larger power nn is larger. If the powers are equal, compare the values of AA.

📐Formulae

a<b (a is less than b)a < b \text{ (a is less than b)}

a>b (a is greater than b)a > b \text{ (a is greater than b)}

ab=a÷b\frac{a}{b} = a \div b

x%=x100x\% = \frac{x}{100}

A×10n (where 1≤A<10)A \times 10^n \text{ (where } 1 \leq A < 10 \text{)}

💡Examples

Problem 1:

Arrange the following in ascending order: 0.70.7, 23\frac{2}{3}, and 68%68\%.

Solution:

Convert all values to decimals: 0.7=0.7000.7 = 0.700 23=0.666…\frac{2}{3} = 0.666\dots 68%=0.68068\% = 0.680 Comparing these decimals: 0.666⋯<0.680<0.7000.666\dots < 0.680 < 0.700 Ascending order: 23,68%,0.7\frac{2}{3}, 68\%, 0.7.

Explanation:

By converting to decimals, we can compare the tenths and hundredths places. 23\frac{2}{3} is approximately 0.670.67, which is smaller than 0.680.68 and 0.70.7.

Problem 2:

Order the following numbers from smallest to largest: −3.4,−3.04,−3.44-3.4, -3.04, -3.44.

Solution:

Consider the position on a number line: −3.44-3.44 is the furthest left. −3.4-3.4 is next. −3.04-3.04 is the closest to zero. Order: −3.44,−3.4,−3.04-3.44, -3.4, -3.04.

Explanation:

In negative numbers, the value with the highest magnitude (3.443.44) is the smallest value because it is furthest from zero on the left side of the number line.

Problem 3:

Arrange in descending order: 4.2×105,4.15×105,0.43×1064.2 \times 10^5, 4.15 \times 10^5, 0.43 \times 10^6.

Solution:

Convert all to the same power of 1010: 4.2×1054.2 \times 10^5 4.15×1054.15 \times 10^5 0.43×106=4.3×1050.43 \times 10^6 = 4.3 \times 10^5 Comparing the coefficients: 4.3>4.2>4.154.3 > 4.2 > 4.15. Descending order: 0.43×106,4.2×105,4.15×1050.43 \times 10^6, 4.2 \times 10^5, 4.15 \times 10^5.

Explanation:

By standardizing the power of 1010 to 10510^5, we can directly compare the decimal parts. 4.34.3 is the largest coefficient, making its original value the largest.

Problem 4:

Find the difference between the largest and smallest number in this set: 12,−5,8,−1212, -5, 8, -12.

Solution:

Identify the values: Largest number =12= 12 Smallest number =−12= -12 Calculation: 12−(−12)24\begin{array}{r} 12 \\ - (-12) \\ \hline 24 \end{array} Difference =24= 24.

Explanation:

Ordering the set gives −12,−5,8,12-12, -5, 8, 12. The difference is calculated as Largest−Smallest\text{Largest} - \text{Smallest}.