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Compare and order rational numbers

Rational-number comparison involves determining relative magnitude and arranging positive and negative fractions, terminating or repeating decimals, and percents, including zero, by interpreting equivalent representations and their positions on a number line. The reasoning accounts for signs, common units or denominators, and absolute value rather than comparing numerators, denominators, or digits in isolation; equivalent forms must have the same value even when their notation differs. This understanding supports ordering values in proportional reasoning, measurement, and algebra and does not extend to irrational numbers or more advanced real-number analysis.

Detailed Explanation: Compare and order rational numbers

To compare and order rational numbers, first rewrite them in the same form. Decimals are often easiest. Then remember:

  • Numbers farther left on a number line are smaller.
  • Any negative number is less than 00 and any positive number.
  • For negative numbers, the one with the greater absolute value is smaller.

Example: Order these numbers from least to greatest:

−35,−55%,0,0.45,12-\frac{3}{5},\quad -55\%,\quad 0,\quad 0.45,\quad \frac{1}{2}

Step 1: Rewrite each number as a decimal.

−35=−0.6-\frac{3}{5}=-0.6 −55%=−0.55-55\%=-0.55 0=00=0 0.45=0.450.45=0.45 12=0.5\frac{1}{2}=0.5

Now compare:

−0.6,−0.55,0,0.45,0.5-0.6,\quad -0.55,\quad 0,\quad 0.45,\quad 0.5

Step 2: Compare the negative numbers.

Both −0.6-0.6 and −0.55-0.55 are negative. Since −0.6-0.6 is farther left on the number line, it is smaller:

−0.6<−0.55-0.6<-0.55

Step 3: Place zero and the positive numbers.

Every negative number is less than 00, and 00 is less than every positive number:

−0.6<−0.55<0<0.45<0.5-0.6<-0.55<0<0.45<0.5

Step 4: Replace the decimals with the original numbers.

−35<−55%<0<0.45<12\boxed{-\frac{3}{5}<-55\%<0<0.45<\frac{1}{2}}

Always compare equivalent values, not just their numerators, denominators, or digits.

Learn by doing: Compare and order rational numbers

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    Number Types (Rational) - Compare - Negative Fractions, Decimals, Repeating Decimals

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