lesson
Different forms can show the same value
1 · Learn
To compare numbers, put them in a common form without changing their values. A terminating decimal is a fraction whose denominator can be a power of ten: 0.375 = 375/1000 = 3/8. Divide numerator and denominator by the same common factor to simplify. Conversely, dividing 3 by 8 gives 0.375.
Percent means per hundred. Multiply a decimal by 100 to express its value as a percentage: 0.375 = 37.5%. This changes the notation, not the amount. To go back, divide the percentage number by 100: 125% = 1.25 = 5/4. Percentages greater than 100% represent more than one whole.
Negative numbers need extra care. On a number line, values increase to the right. -0.8 is to the LEFT of -3/4 = -0.75, so -0.8 < -3/4. A larger magnitude below zero is a smaller number. The symbol < means less than, > greater than, and ≠ not equal. The symbols ≤ and ≥ also allow equality.
Compare -0.8, -3/4, 70% and 5/4 by converting: -0.8, -0.75, 0.7, 1.25. Their ascending order is -0.8 < -3/4 < 70% < 5/4. Do not rank them just by the printed digits 8, 3, 70 and 5.
When comparing quantities, say which one is the reference whole and match their units. A length of 36 cm is 36/24 = 3/2 of a 24 cm reference length, or 150%. Reversing the comparison gives 24/36 = 2/3, not 150%. Percent as an operator works the other way: 150% of 24 cm is 1.5×24 = 36 cm.
2 · Worked example
Compare -0.6 and -5/8. Since 5÷8 = 0.625, the second is -0.625. Thus -5/8 < -0.6. Separately, 45 cm as a percentage of 36 cm is (45/36)×100% = 125%. Both quantities use centimetres.
3 · Your turn
Write 0.875 as a simplified fraction and percentage. Put -0.7, -3/4, 60% and 7/5 in ascending order. Express 56 cm as a percentage of 40 cm.
Check your answer
7/8 and 87.5%. Ascending: -3/4 < -0.7 < 60% < 7/5. The length comparison is 56/40 = 1.4, so 140%.
4 · Apply your learning
Draw a number line from -1 to 1.5 on paper or the working pad. Mark all four practice values. Explain the negative pair using left and right, then show 7/5 as one whole and two fifths.
Adult guidance and safety
Ask pupils to identify the reference whole and explain each equivalent representation. Keep answers exact until rounding is explicitly requested. Use counters or a bar diagram for sharing before a symbolic rule.
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Check a calculation · Year 5 onward
Use this only when your teacher or adult allows a calculator for the task. Show your thinking first. It does not replace mental or written arithmetic and does not mark your lesson answer.
Powers, roots and trigonometry · Year 7 onward
Use only when the task allows a calculator. Choose one operation at a time and record your working. Trigonometry here always uses degrees (DEG), not radians. Inverse sine, cosine and tangent return an angle, not a reciprocal. Keep exact fractions and π in answers when requested.
Approximate numerical results use up to 12 significant digits. Inverse sine returns −90° to 90°, inverse cosine 0° to 180°, and inverse tangent −90° to 90°; these are principal values, not every possible solution of a trigonometric equation.
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