Ratios, Rates, Percentages, and Units flashcards

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Changing a ratioRatios and Proportional Relationships: Write the before counts first and update only the category that changes.
When should I use allocate a total?Ratios and Proportional Relationships: Use a part-to-part ratio to divide a known total.
What method supports changing a ratio?Ratios and Proportional Relationships: a k/(b k + d)=new ratio
Direct proportionRatios and Proportional Relationships: Divide once for the per-unit multiplier, then multiply for a new input.
Allocate a totalRatios and Proportional Relationships: Find one ratio part, then scale the requested group.
What must I check in ratios and proportional relationships?Ratios and Proportional Relationships: Maintain ratio order and distinguish proportional models from those with fixed offsets
What method supports direct proportion?Ratios and Proportional Relationships: y=kx, k=y/x for x≠0
What method supports allocate a total?Ratios and Proportional Relationships: first count = total·a/(a+b)
When should I use changing a ratio?Ratios and Proportional Relationships: Model a change to one category before imposing a new ratio.
When should I use direct proportion?Ratios and Proportional Relationships: Recover the constant of proportionality and predict an output.
Combined work ratesUnit Rates, Speed, and Combined Quantities: Add work per hour, then invert to get hours per task.
When should I use unit rate?Unit Rates, Speed, and Combined Quantities: Find a per-unit amount and scale it.
What method supports combined work rates?Unit Rates, Speed, and Combined Quantities: 1/T=1/a+1/b
Average speedUnit Rates, Speed, and Combined Quantities: Add the journey first; divide once at the end.
Unit rateUnit Rates, Speed, and Combined Quantities: Find how much belongs to one unit.
What must I check in unit rates, speed, and combined quantities?Unit Rates, Speed, and Combined Quantities: Average speed is total distance divided by total time, not generally the mean of component speeds
What method supports average speed?Unit Rates, Speed, and Combined Quantities: average speed=(d₁+d₂)/(t₁+t₂)
What method supports unit rate?Unit Rates, Speed, and Combined Quantities: rate=quantity/time; quantity=rate×time
When should I use combined work rates?Unit Rates, Speed, and Combined Quantities: Add rates when independent workers contribute to the same total.
When should I use average speed?Unit Rates, Speed, and Combined Quantities: Combine distances and times before computing an overall rate.
Area and volume unitsUnit Conversions and Dimensional Analysis: Square the length conversion for area and cube it for volume.
When should I use single-unit conversion?Unit Conversions and Dimensional Analysis: Cancel a starting unit using a conversion ratio.
What method supports area and volume units?Unit Conversions and Dimensional Analysis: 1 m²=10,000 cm²; 1 m³=1,000,000 cm³
Rate conversionsUnit Conversions and Dimensional Analysis: Cancel distance and time units separately.
Single-unit conversionUnit Conversions and Dimensional Analysis: Put the old unit in the denominator so it cancels.
What must I check in unit conversions and dimensional analysis?Unit Conversions and Dimensional Analysis: Use stated conversion factors exactly; squared and cubed units require squared and cubed factors
What method supports rate conversions?Unit Conversions and Dimensional Analysis: km/h×(1000 m/km)×(1 h/3600 s)=m/s
What method supports single-unit conversion?Unit Conversions and Dimensional Analysis: value in new units=value in old units×conversion factor
When should I use area and volume units?Unit Conversions and Dimensional Analysis: Raise conversion factors to match the dimension.
When should I use rate conversions?Unit Conversions and Dimensional Analysis: Convert both numerator and denominator units.
Geometric density modelsDensity and Other Derived Units: Find the space occupied, then apply mass per unit space.
When should I use density as a quotient?Density and Other Derived Units: Calculate density from mass and volume.
What method supports geometric density models?Density and Other Derived Units: rectangular-solid mass=ρ·lwh
Derived-unit modelingDensity and Other Derived Units: Multiply a per-unit quantity by the matching number of units.
Density as a quotientDensity and Other Derived Units: Divide how much mass there is by how much space it occupies.
What must I check in density and other derived units?Density and Other Derived Units: Match the mass and volume units to the requested density; density is an intensive ratio
What method supports derived-unit modeling?Density and Other Derived Units: m=ρV; population=(people/area)×area; cost=($/kWh)×kWh
What method supports density as a quotient?Density and Other Derived Units: ρ=m/V, V>0
When should I use geometric density models?Density and Other Derived Units: Combine a volume formula with density and unit consistency.
When should I use derived-unit modeling?Density and Other Derived Units: Use a derived unit to recover a total amount.
Several charges with stated basesPercentages, Discounts, Markups, Taxes, and Tips: Label each percent base before calculating.
When should I use discounted price?Percentages, Discounts, Markups, Taxes, and Tips: Multiply by the retained-price factor.
What method supports several charges with stated bases?Percentages, Discounts, Markups, Taxes, and Tips: total=base+tax-base×tax rate+tip-base×tip rate
Markup and taxPercentages, Discounts, Markups, Taxes, and Tips: An added percentage means retain the whole and add the extra part.
Discounted pricePercentages, Discounts, Markups, Taxes, and Tips: Keep the percentage that remains after the discount.
What must I check in percentages, discounts, markups, taxes, and tips?Percentages, Discounts, Markups, Taxes, and Tips: Apply each rate to its stated base; round currency only at the requested final stage
What method supports markup and tax?Percentages, Discounts, Markups, Taxes, and Tips: total=base×(1+r)
What method supports discounted price?Percentages, Discounts, Markups, Taxes, and Tips: sale price=original×(1−discount rate)
When should I use several charges with stated bases?Percentages, Discounts, Markups, Taxes, and Tips: Apply tax and tip to the specified amounts.
When should I use markup and tax?Percentages, Discounts, Markups, Taxes, and Tips: Use an increase factor for a stated base.
Net successive changePercent Change, Reverse Percentages, and Successive Changes: Multiply the factors first and convert the product back to a rate.
When should I use percent change?Percent Change, Reverse Percentages, and Successive Changes: Compute an increase or decrease relative to the original.
What method supports net successive change?Percent Change, Reverse Percentages, and Successive Changes: net rate=(1+r)(1−s)−1
Reverse percentagesPercent Change, Reverse Percentages, and Successive Changes: Undo the multiplier with division.
Percent changePercent Change, Reverse Percentages, and Successive Changes: Compare the change with what you started with.
What must I check in percent change, reverse percentages, and successive changes?Percent Change, Reverse Percentages, and Successive Changes: Percent change uses the original amount as its denominator; reverse a percent by division
What method supports reverse percentages?Percent Change, Reverse Percentages, and Successive Changes: original=final/(1+r) or final/(1−r)
What method supports percent change?Percent Change, Reverse Percentages, and Successive Changes: percent change=100(final−initial)/initial
When should I use net successive change?Percent Change, Reverse Percentages, and Successive Changes: Compute one net percent from a product of factors.
When should I use reverse percentages?Percent Change, Reverse Percentages, and Successive Changes: Recover an original amount from a changed amount.
Area from a length scaleScale Drawings and Contextual Proportions: Area scales along both length and width.
When should I use map distance?Scale Drawings and Contextual Proportions: Apply a map scale with units.
What method supports area from a length scale?Scale Drawings and Contextual Proportions: area factor=(length factor)²
Recover a drawing dimensionScale Drawings and Contextual Proportions: Divide the real measurement by the amount represented by one drawing unit.
Map distanceScale Drawings and Contextual Proportions: Drawing units and real-world units are different quantities.
What must I check in scale drawings and contextual proportions?Scale Drawings and Contextual Proportions: Use the scale in the stated direction and square a length scale only when converting areas
What method supports recover a drawing dimension?Scale Drawings and Contextual Proportions: drawing length=actual length/scale rate
What method supports map distance?Scale Drawings and Contextual Proportions: actual distance=drawing distance×actual units per drawing unit
When should I use area from a length scale?Scale Drawings and Contextual Proportions: Convert a drawing area by squaring the scale factor.
When should I use recover a drawing dimension?Scale Drawings and Contextual Proportions: Divide by the actual-to-drawing scale.