Goal and prerequisites
Carry out operations in different number representations while preserving their values. Review proportion before starting.
Method and its conditions
In standard form multiply coefficients and combine powers, then normalize the coefficient into [1,10). To convert a recurring decimal, align identical repeating tails by multiplying by a power of ten and subtract. For repeated percentage change, multiply the successive factors.
Worked example: show the decisions
(3×10⁴)(5×10⁻²)=15×10²=1.5×10³. For x=0.272727..., subtract x from 100 x: 99 x=27, giving x=3/11. A balance of 200 growing by 10% twice becomes 200×1.1²=242.
Contrast: diagnose an incorrect approach
The intermediate coefficient 15 is not normalized standard form, though the value is correct. A recurring decimal is not its truncated two-decimal approximation. Two increases of 10% give a total increase of 21%, because the second increase applies to the new amount.
Transfer: try a changed situation
A quantity becomes 363 after two increases of 10%. Find its original value. Also write (6×10⁵)/(2×10²) in standard form.
Transfer solution and interpretation
The original is 363/1.21=300. The quotient is 3×10³: divide coefficients and subtract exponents. Both answers can be checked by applying the forward operations.
Write (3×10³)(5×10⁻²) as c×10² in standard form. Find c.
- Multiply coefficients and add exponents, then normalize the coefficient for the specified power 10².
- Using the stated quantities: 3×5/10 = 3/2.
- Check the domain, units and whether the answer is an exact value, bound or estimate.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
0580 syllabus for exams in 2025–2027