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Math: Ratios and Proportional Relationships

Unit rate, the constant of proportionality, and the whole percent family — tip, tax, discount, markup, commission, interest and error — plus unit conversion.

16 topics · 66 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

Ratios and Equivalent Ratios

  1. How it worksHow does scaling both parts of a ratio by the same factor leave the ratio unchanged?
  2. How it worksHow does a part-to-part ratio get rewritten as a part-to-whole fraction?
  3. Worked exampleFind how many girls sit in a class of 30 when boys to girls run 2 to 3.
  4. Common mistakeWhy isn't a 2 to 3 ratio of boys to girls the same as saying 2/3 of the class are boys?
  5. Going furtherWhat happens to a ratio problem when the total is hidden and only the difference between parts is given?
Your turneasy1 of 3
In a class the ratio of boys to girls is 7 to 3. The class has n students. Write an expression for the number of girls, in terms of n.

Unit Rate and Rates Built from Fractions

  1. How it worksHow does dividing one quantity by the other squash a rate down to a single per-one number?
  2. Worked exampleFind the better buy between 12 ounces for 4.20 dollars and 20 ounces for 6.60 dollars.
  3. Worked exampleFind the speed in miles per hour when a runner covers 3/4 mile in 1/6 of an hour.
  4. Common mistakeWhy doesn't the cheaper package win automatically once the two sizes stop matching?
Your turnmedium1 of 3
A runner covers 3/4 of a mile in 1/10 of an hour. What is the speed in miles per hour?

Proportional Relationships and the Constant of Proportionality

  1. How it worksHow does one constant k show up as the same number in a table, on a graph and in an equation?
  2. How it worksHow does the equation y = kx capture an entire proportional relationship with a single number?
  3. Worked exampleFind the constant of proportionality when 7 notebooks cost 10.15 dollars, then write the equation.
  4. Going furtherWhat happens to the table, the graph and the equation all at once when k gets larger?
Your turneasy1 of 3
Seven notebooks cost 23.51 dollars. What is the constant of proportionality, in dollars per notebook?

Recognizing Proportionality in a Table, Graph, or Equation

  1. How it worksHow do you test a table for proportionality by checking every y divided by x for one shared value?
  2. How it worksHow does the starting value of a line decide whether it is proportional or merely linear?
  3. Common mistakeWhy isn't every straight line a proportional relationship when each one climbs at a steady rate?
  4. Worked exampleFind which of y = 4x, y = 4x + 2 and xy = 4 is proportional and show how the other two fail.
Your turneasy1 of 3
Let 7 be a positive number. Three relationships are given: y = 7x y = 7x + 2 xy = 7 Which one is proportional?

Reading Points on a Proportional Graph

  1. How it worksHow does the origin on a proportional graph say that zero items cost zero dollars?
  2. How it worksHow does any point on the line translate back into a full sentence about the situation?
  3. Worked exampleFind what the point (4, 30) means on a cost versus pounds graph, and pull the unit rate from it.
Your turneasy1 of 3
On a graph of cost against pounds, the point (8, c) lies on a line through the origin. Write an expression for the cost of one pound, in dollars, in terms of c.

Setting Up and Solving Proportions

  1. How it worksHow does cross multiplication collapse a proportion into a one-step equation?
  2. Worked exampleFind x when 3/8 equals x/52, then check by comparing the two ratios side by side.
  3. Worked exampleFind how far 15 gallons carry a car that travels 96 miles on 4 gallons.
  4. Going furtherWhat happens to the setup when adding workers makes the time go down instead of up?
Your turneasy1 of 3
Solve 5/8 = x/d for x, writing your answer in terms of d.

Percent of a Number

  1. How it worksHow does rewriting 15 percent as 0.15 make a percent-of question one multiplication long?
  2. Worked exampleFind 35 percent of 240 and then 8 percent of 62.50 without a long multiplication.
  3. Common mistakeWhy isn't asking for 150 percent of a number a nonsense request?
Your turneasy1 of 3
Find 35 percent of 315.

Finding the Whole from a Percent

  1. How it worksHow does part equals percent times whole let you solve for the whole instead of the part?
  2. Worked exampleFind the original number when 18 is 30 percent of it, then check by taking 30 percent back.
  3. Worked exampleFind the list price when a 25 percent discount left a shopper paying 96 dollars.
  4. Common mistakeWhy isn't the whole always larger than the part once the percent climbs past 100?
Your turneasy1 of 3
The number 45 is 30 percent of some number. What is that number?

Percent Increase and Percent Decrease

  1. How it worksHow does the change divided by the original amount become a percent increase or decrease?
  2. How it worksHow does multiplying by 1.05 do the same job as adding 5 percent of a number to itself?
  3. Worked exampleFind the percent increase when rent climbs from 1250 dollars to 1400 dollars.
  4. Worked exampleFind the new population after a town of 4800 people shrinks by 15 percent.
  5. Common mistakeWhy isn't the percent change the same size when you swap which number counts as the original?
Your turneasy1 of 3
Rent climbs from 1250 dollars to 1509 dollars. Find the percent increase.

Successive Percent Changes

  1. How it worksHow does chaining multipliers such as 1.1 times 1.1 replace adding the percents together?
  2. Worked exampleFind the final price of a 60 dollar jacket marked down 25 percent and then another 20 percent.
  3. Worked exampleFind the one percent change equal to a 30 percent rise followed by a 30 percent fall.
  4. Common mistakeWhy doesn't a 40 percent loss get undone by a 40 percent gain the following year?
  5. Going furtherWhat happens to the final amount when the two percent changes are applied in the other order?
Your turneasy1 of 3
A jacket costing 96 dollars is marked down 25 percent and then marked down another 20 percent from the sale price. What is the final price, in dollars?

Tax, Tip, and Discount

  1. How it worksHow does a single multiplier handle a discount and a tax in one step rather than two?
  2. Worked exampleFind the total on a 78 dollar meal carrying a 15 percent tip and 8.875 percent sales tax.
  3. Worked exampleFind the sale price of a 145 dollar coat at 30 percent off before any tax gets added.
  4. Common mistakeWhy doesn't applying the tax before the discount change what the customer finally pays?
Your turnmedium1 of 3
A meal costs 96 dollars. A 15 percent tip and 8.875 percent sales tax are both figured on the meal price. What is the total, in dollars?

Markup and Commission

  1. How it worksHow does a markup measured against cost differ from a margin measured against the selling price?
  2. Worked exampleFind the selling price when a shop buys a bike for 180 dollars and marks it up 45 percent.
  3. Worked exampleFind total pay on 12400 dollars of sales at 6 percent commission plus a 400 dollar base.
  4. Going furtherWhat happens to the price when a 20 percent markup is followed by a 20 percent markdown?
Your turneasy1 of 3
A shop buys a bike for 245 dollars and marks it up 45 percent. What is the selling price, in dollars?

Simple Interest

  1. How it worksHow does I = prt turn a rate, a balance and a stretch of time into one interest amount?
  2. Worked exampleFind the interest earned on 2400 dollars at 5.5 percent simple interest over 3 years.
  3. Worked exampleFind how long 1500 dollars must sit at 4 percent simple interest to earn 300 dollars.
  4. Common mistakeWhy doesn't simple interest match what a real savings account pays after several years?
Your turneasy1 of 3
Write an expression for the simple interest earned on P dollars at 6 percent per year over 5 years, in terms of P.

Percent Error

  1. How it worksHow does dividing the size of the miss by the true value turn an error into a percent?
  2. Worked exampleFind the percent error when a student measures 47 centimeters and the true length is 50.
  3. Common mistakeWhy doesn't percent error care whether the measurement came in high or low?
Your turneasy1 of 3
A student measures a length as 46 centimeters. The true length is 50 centimeters. Find the percent error.

Unit Conversion and Multi-Step Exchange

  1. How it worksHow does a conversion factor written as a fraction cancel the unit you want gone?
  2. How it worksHow do you chain two conversion factors when no single step links the units you have to the ones you want?
  3. Worked exampleFind the dollars Malik receives for 140 lorgs and 16 dalts when 1 dollar is 7 lorgs and 0.5 dalt.
  4. Worked exampleFind how many feet per second a car covers while travelling 45 miles per hour.
  5. Common mistakeWhy doesn't a conversion factor flipped upside down still deliver the right answer?
Your turnmedium1 of 3
A car travels 56 miles per hour. How many feet per second is that? Use 5280 feet in a mile and 3600 seconds in an hour.

Multi-Step Rate Problems

  1. How it worksHow does writing a rate problem one step at a time keep the units from tangling together?
  2. Worked exampleFind the average speed over 120 miles at 40 miles per hour then 120 miles at 60.
  3. Worked exampleFind the hours a printer running 18 pages a minute needs to finish a 4860 page job.
  4. Common mistakeWhy doesn't averaging two speeds give the average speed of a round trip?
  5. Ties togetherHow does a steady rate problem turn into the same y = kx line you graph for proportions?
Your turnhard1 of 3
A driver covers 120 miles at 44 miles per hour and then another 120 miles at v miles per hour. Write an expression for the average speed for the whole trip, in terms of v.