Examkin

Question

Difficulty: MediumFractions and Decimals

An agronomist is comparing the soil moisture retention rates of four different agricultural plots. The rates, measured in liters per cubic meter per day, are given below. Arrange the plots in ascending order of their moisture retention rates.

  1. 1Plot Beta: 139\frac{13}{9}
  2. 2Plot Delta: 1.451.45
  3. 3Plot Gamma: 1.45‾1.\overline{45}
  4. 4Plot Alpha: 1.45‾1.4\overline{5}

Answer

The correct ascending order is Plot Beta, Plot Delta, Plot Gamma, and then Plot Alpha.
By converting all rates to extended decimals, we get: Plot Beta (1.444...1.444...), Plot Delta (1.4501.450), Plot Gamma (1.4545...1.4545...), and Plot Alpha (1.4555...1.4555...). Comparing them digit-by-digit from left to right establishes the strictly ascending order: Beta, Delta, Gamma, Alpha.

Step-by-Step Solution

1
Convert the fraction into a decimal to create a uniform format for comparison.
Plot Beta: 139=1.4444...\frac{13}{9} = 1.4444...
Decimal forms allow for direct digit-by-digit comparison of magnitudes.
2
Expand all values to at least four decimal places to clearly see the repeating patterns.
Plot Alpha: 1.4555...1.4555...
Plot Beta: 1.4444...1.4444...
Plot Gamma: 1.4545...1.4545...
Plot Delta: 1.4500...1.4500...
Truncating or estimating too early can hide the subtle differences between terminating and repeating decimals.
3
Compare the tenths and hundredths places.
All values have a 44 in the tenths place. In the hundredths place, Plot Beta has a 44, while the others have a 55. Thus, Plot Beta is the smallest.
Comparing digits from left to right establishes the overall magnitude.
4
Compare the thousandths place for the remaining three plots.
Plot Delta has 00, Plot Gamma has 44, and Plot Alpha has 55.
This establishes the final order for the remaining values: Delta (1.450...1.450...) < Gamma (1.4545...1.4545...) < Alpha (1.4555...1.4555...).

Key Concept

Comparing terminating decimals, recurring decimals, and fractions by converting them to expanded decimal forms.
Rate this question

Topics

Fractions and Decimals

More questions from this topic