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Understanding Ratios, Proportions, and More in Mathematics

Explore the concepts of ratios, proportions, direct and inverse proportions, density, and percentages in mathematics. Learn how to calculate ratios, solve proportion problems, and understand the relevance of density in practical applications. Practice converting percentages to decimals and vice versa, and understand the concept of percent error in measurements.

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Understanding Ratios, Proportions, and More in Mathematics

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  1. Ratios, Proportions, and More… 3221.MATH.1 3221.MATH.2

  2. Ratio The relative size of two quantities expressed as the quotient of one divided by the other The ratio of a to b is a:b or a/b

  3. Proportions An equality between ratios resulting in a constant

  4. Directly Proportional Dividing one value by the other gives a constant value, k y = k x This equation can be written as follows: y = k x As y increases, x increases. As y decreases, x decreases. (2 properties increase or decrease together)

  5. Direct Proportions M1 =M2 V1V2

  6. Inversely Proportional Multiplying one value by another gives a constant value, k xy = k As y increases, x decreases. As y decreases, x increases. The graph of this is a hyperbola.

  7. Inverse Proportions P1V1 = P2V2

  8. Percent % Value that gives the ratio to the whole Per centum “1 out of 100” 22% “22 out of 100” 22/100 = .22

  9. Density Ratio of mass to volume (amount of a substance in a given volume) D = m v the density of water = 1.00 g/cm3 Units of Volume: cm3, mL, m3, dm3, L Thus, the units of density may be expressed as: (g, kg) . (cm3, mL, dm3, L)

  10. Why is density relevant? The density of substances is an important consideration in choosing materials from which to make things. Airplanes, for example, are made of low-density materials, while bridges are made of higher-density materials. Density is also important as a conversion factor between mass and volume, and vice versa.

  11. Find the density of a substance that has a mass of 31 g and a volume of 68 cm3.

  12. Diamond has a density of 3.26 g/ cm3. What is the mass of a diamond that has the volume of 0.351 cm3?

  13. The mass of fuel in an airplane must be carefully accounted for before takeoff. If a 747 contains 155,211 L of fuel, what is the mass of the fuel in kg? Assume the density of the fuel to be 0.768 g/cm3.

  14. Acetone (fingernail-polish remover) has a density of 0.7857 g/cm3.a. What is the mass in grams of 28.56 mL of acetone?b. What is the volume in milliliters of 6.54 g of acetone?

  15. Working with percentages • Mechanics of changing a percentage to a decimal: move decimal 2 places to left • The percentage symbol helps remind you of this %

  16. Practice • What is 20% of 35? • 11 is what percent of 20? • 6.5% of what is 80.02? • What is 10 after being increased by 15%? • What is 201.1 after being decreased by 25%?

  17. Percent Error Actual-Measuredx 100 Actual

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