Math Problem Statement

A series of conversion and calculation problems involving time, length, mass, and scientific notation. The user is tasked with solving these unit conversion questions, often using powers of ten.

Solution

Let's go through each of the questions in the image and solve them step by step.

1. 0.02 jam+20 menit=  detik0.02 \, \text{jam} + 20 \, \text{menit} = \, \, \text{detik}

  • 1 hour = 3600 seconds, so:
    • 0.02 jam=0.02×3600=72 detik0.02 \, \text{jam} = 0.02 \times 3600 = 72 \, \text{detik}
  • 1 minute = 60 seconds, so:
    • 20 menit=20×60=1200 detik20 \, \text{menit} = 20 \times 60 = 1200 \, \text{detik}

Thus, 72+1200=1272 detik72 + 1200 = 1272 \, \text{detik}

2. 12 menit+10 detik=  jam\frac{1}{2} \, \text{menit} + 10 \, \text{detik} = \, \, \text{jam}

  • 12 menit=30 detik\frac{1}{2} \, \text{menit} = 30 \, \text{detik}
  • Total seconds = 30+10=40 detik30 + 10 = 40 \, \text{detik}
  • 1 hour = 3600 seconds, so: 40 detik=403600 jam=190 jam40 \, \text{detik} = \frac{40}{3600} \, \text{jam} = \frac{1}{90} \, \text{jam} or approximately 0.0111 jam0.0111 \, \text{jam}.

3. 0.002 hm+0.01 dam=  km0.002 \, \text{hm} + 0.01 \, \text{dam} = \, \, \text{km}

  • 1 hectometer (hm) = 0.1 kilometers (km), so: 0.002 hm=0.002×0.1=0.0002 km0.002 \, \text{hm} = 0.002 \times 0.1 = 0.0002 \, \text{km}
  • 1 dekameter (dam) = 0.01 kilometers (km), so: 0.01 dam=0.01×0.01=0.0001 km0.01 \, \text{dam} = 0.01 \times 0.01 = 0.0001 \, \text{km}

Thus, 0.0002+0.0001=0.0003 km0.0002 + 0.0001 = 0.0003 \, \text{km}

4. 10×10−2 m+4×10−4 dm=  cm10 \times 10^{-2} \, \text{m} + 4 \times 10^{-4} \, \text{dm} = \, \, \text{cm}

  • 10×10−2 m=0.1 m=10 cm10 \times 10^{-2} \, \text{m} = 0.1 \, \text{m} = 10 \, \text{cm} (since 1 m = 100 cm)
  • 4×10−4 dm=4×10−4×10 cm=0.0004×10=0.004 cm4 \times 10^{-4} \, \text{dm} = 4 \times 10^{-4} \times 10 \, \text{cm} = 0.0004 \times 10 = 0.004 \, \text{cm}

Thus, 10+0.004=10.004 cm10 + 0.004 = 10.004 \, \text{cm}

5. 0.0004 kg+10−4 kg=  g0.0004 \, \text{kg} + 10^{-4} \, \text{kg} = \, \, \text{g}

  • 0.0004 kg=0.0004×1000=0.4 g0.0004 \, \text{kg} = 0.0004 \times 1000 = 0.4 \, \text{g}
  • 10−4 kg=10−4×1000=0.1 g10^{-4} \, \text{kg} = 10^{-4} \times 1000 = 0.1 \, \text{g}

Thus, 0.4+0.1=0.5 g0.4 + 0.1 = 0.5 \, \text{g}

6. 4.5×10−4÷2×10−6=  4.5 \times 10^{-4} \div 2 \times 10^{-6} = \, \,

This is a simple division of scientific notation: 4.5×10−42×10−6=2.25×102=225\frac{4.5 \times 10^{-4}}{2 \times 10^{-6}} = 2.25 \times 10^{2} = 225

7. 0.00125×10−2=0.001×10−9×10−5=  0.00125 \times 10^{-2} = 0.001 \times 10^{-9} \times 10^{-5} = \, \,

  • Simplifying both sides: 0.00125×10−2=1.25×10−50.00125 \times 10^{-2} = 1.25 \times 10^{-5} and 0.001×10−9×10−5=10−3×10−9×10−5=10−170.001 \times 10^{-9} \times 10^{-5} = 10^{-3} \times 10^{-9} \times 10^{-5} = 10^{-17} Thus, 1.25×10−5=1.25×10−171.25 \times 10^{-5} = 1.25 \times 10^{-17}

8. 2×10−9 jam−4×10−2 menit=  2 \times 10^{-9} \, \text{jam} - 4 \times 10^{-2} \, \text{menit} = \, \,

  • 4×10−2 menit=4×10−2×160 jam=6.67×10−4 jam4 \times 10^{-2} \, \text{menit} = 4 \times 10^{-2} \times \frac{1}{60} \, \text{jam} = 6.67 \times 10^{-4} \, \text{jam}
  • Subtracting the two: 2×10−9−6.67×10−4≈−6.67×10−4 jam2 \times 10^{-9} - 6.67 \times 10^{-4} \approx -6.67 \times 10^{-4} \, \text{jam}

9. 2×10−2×4×10−4÷2×10−5=  2 \times 10^{-2} \times 4 \times 10^{-4} \div 2 \times 10^{-5} = \, \,

First, multiply the terms in the numerator: 2×4=8and10−2×10−4=10−62 \times 4 = 8 \quad \text{and} \quad 10^{-2} \times 10^{-4} = 10^{-6} Thus, 8×10−62×10−5=4×10−1=0.4\frac{8 \times 10^{-6}}{2 \times 10^{-5}} = 4 \times 10^{-1} = 0.4

Would you like more details on any of the steps?

Here are five related questions:

  1. How do you handle complex unit conversions involving scientific notation?
  2. What is the process for converting between different time units, such as minutes to hours?
  3. How do you simplify expressions with powers of 10 in scientific notation?
  4. What are some common conversions between metric length units (e.g., meters to centimeters)?
  5. Can you explain how to manipulate and calculate with small values in scientific notation?

Tip: When working with scientific notation, always align powers of 10 and use rules of exponents for easier multiplication or division.

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Math Problem Analysis

Mathematical Concepts

Unit Conversion
Time Conversion
Metric System
Scientific Notation
Division and Multiplication with Powers of 10

Formulas

1 hour = 3600 seconds
1 minute = 60 seconds
1 hectometer (hm) = 0.1 kilometers (km)
1 dekameter (dam) = 0.01 kilometers (km)
1 meter = 100 centimeters
1 kilogram = 1000 grams

Theorems

Rules of Exponents
Metric Conversion

Suitable Grade Level

Grades 9-12

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