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Difficulty: MediumFractions and Decimals

An aerospace engineering facility is calibrating a pressure sensor using four specialized metallic foils. To prevent calibration errors, a technician must stack these foils in strictly ascending order of their thickness (from thinnest to thickest).

The thicknesses of the four foils are:
- Foil P: 0.3450.345 mm
- Foil Q: 1132\frac{11}{32} mm
- Foil R: 0.34‾0.\overline{34} mm
- Foil S: 720\frac{7}{20} mm

Arrange the foils in the correct ascending order of their thickness.

  1. 1Foil R (0.34‾0.\overline{34} mm)
  2. 2Foil Q (1132\frac{11}{32} mm)
  3. 3Foil P (0.3450.345 mm)
  4. 4Foil S (720\frac{7}{20} mm)

Answer

The correct ascending sequence from thinnest to thickest is Foil R, Foil Q, Foil P, and Foil S.
By converting all foil thicknesses into uniform decimal representations, we establish the values: Foil R (0.3434...0.3434...), Foil Q (0.343750.34375), Foil P (0.3450.345), and Foil S (0.350.35). Comparing these exact decimal values digit-by-digit confirms that the ascending sequence must be R, Q, P, then S.

Step-by-Step Solution

1
Convert the fraction for Foil S to a decimal.
720=7×520×5=35100=0.35\frac{7}{20} = \frac{7 \times 5}{20 \times 5} = \frac{35}{100} = 0.35 mm
Converting all measurements to a uniform decimal format allows for direct magnitude comparison.
2
Convert the fraction for Foil Q to a decimal.
11÷32=0.3437511 \div 32 = 0.34375 mm
Using long division provides the exact decimal expansion necessary to compare against closely grouped measurements.
3
Expand the recurring decimal for Foil R.
0.34‾=0.343434...0.\overline{34} = 0.343434... mm
Expanding the repeating digits to at least 4 decimal places makes place-value comparison clear.
4
Compare all four decimal values place-by-place and sequence them from smallest to largest.
0.343434...<0.34375<0.34500<0.350000.343434... < 0.34375 < 0.34500 < 0.35000
Aligning the decimals by their thousandths and ten-thousandths places reveals the exact ascending sequence.

Key Concept

Converting fractions and recurring decimals into standard decimals to accurately evaluate and compare their magnitudes.
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Topics

Fractions and Decimals

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