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Word Problems

Browse Middle School Ratio/Proportion
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:
    Mixture problems.
    Unit conversions, unit cancellation.
    What is a ratio?

Converting Units across Measurements [08/31/2010]
How do you convert units to a different measurement? And how do you know how many decimal places to move as a result? Doctor Ian suggests making the multiplication of units explicit, and introduces ratios of two quantities equal to each other.

Cross Products [02/28/1999]
I am having trouble understanding cross-products.

Currency Exchange Rates [01/02/1999]
Do you know how much half a crown, a crown, and a shilling are worth in American money?

Defining Counts [07/05/2016]
An adult struggles to disambiguate a unit label from the variable that represents the amount of that unit. Doctor Peterson clears up the confusion before reconciling the different uses with explicit labeling and dimensional analysis.

Degree Measure of a Central Angle [07/29/1997]
Find the degree measure, to the nearest tenth, of a central angle whose intercepted arc measures 21 cm. in a circle of radius 4 cm.

The Difference between Calculating Markup and Profit Margin [08/11/2006]
If we have fixed costs of $50,000 on a product and want to have a profit margin of 25%, how much should we charge the customer to buy it?

Difference in Radius, Difference in Speed [06/25/2002]
What is the difference in speed between a point on the tread of a tire and another point closer to the center, if the tire is rotating at 5000 rpm?

Distances From a Point on a Line [3/21/1995]
A student put points P and Q on a line to the right of point X. Three times the distance from point X to point P is four times the distance from point X to point Q. What is the ratio of PQ:XQ? The answer key says 1:4.......the class and teacher disagree. What do you say?

Does Unit Price Always Give the Best Buy? [12/30/2007]
Which is the best buy, 13 videos for $25 or 11 videos for $20? It seems like there are a couple ways to analyze the question.

Dog Food Mixture [10/23/2002]
A grocer wishes to mix 6 pounds of Chippy dog food worth $3 per pound with Hippy dog food, worth $4 per pound. How much of the Hippy dog food should be mixed if the dog food mixture is to sell for $3.20?

Elephant Feet [05/16/1997]
Why do elephants have big feet?

Estate Distribution [05/10/1999]
Does 1/10 of the estate go to the church before or after taxes?

Evaporation [12/08/2003]
A substance is 99% water. Some water eveporates, leaving a substance that is 98% water. How much of the water evaporated?

Explaining Proportions and Unit Prices [09/21/2006]
If you need 5 lbs of flour to make 10 lbs of batter, how many pounds of flour do you need for 1 lb of batter? I know how to do this problem (divide 5 by 10) but I don't understand why dividing the 5 lbs by the whole (10 lbs) gives you the amount for a SINGLE pound of batter.

Finding Original Amount before a Percent Was Added [04/24/2007]
If I paid $25.00 for gas and that includes 42% in taxes, what was the cost of the gas before the taxes?

A Formula for the Scaling Factor of a Constantly Decreasing Image [9/28/1995]
In a picture of a person holding a mirror and looking into a mirror, the image is repeated over and over again. The large picture is made of an infinite number of smaller pictures, all of which are proportional. The first image is about 15cm. The second image is about 5.8cm. What is the actual height of the 10th image in the picture?

Four Methods of Solving Classic Chicken and Egg Problems [02/24/2009]
If 6 cats can kill 6 rats in 6 minutes, how many cats will it take to kill 100 rats in 50 minutes? A look at four different solution methods for this sort of problem.

Fractions as Ratios: Wholly or Partially? [03/09/2014]
An adult wonders whether to classify fractions as ratios. Doctor Peterson puts the terms in perspective.

Height/Weight Ratio [05/15/1997]
You have a friend who weighs 155 pounds, but is only 5 feet 4 inches tall. What is the height-to-weight ratio of your pal?

History of Ratio and Proportion [12/28/2003]
Who discovered ratios and proportions? When did it happen?

Lemonade Proportions [04/02/1998]
If lemonade is made with a ratio of 3 cups water to 2 cups lemon juice, how many cups of lemon juice are in 10 gallons of lemonade?

Limits to Paper Folding [05/09/2002]
What is the limit to the number of times that a piece of paper can be folded, and how is it established?

Maps and Square Feet [05/24/1999]
If 1 inch = 50 feet, what is the area of a location that is 10000 square feet?

Math and Music: Harmonic Series [01/09/1998]
Could you supply me with information on how math relates to music?

Math and the Piano [09/27/1998]
How is math is related to the piano? What about tempo, pitch, and tuning?

Math Fair Project Ideas [11/14/1995]
I am a 7th grade student. Our school is having a Math Fair. Can you suggest an interesting project using algebra and/or geometry?

Mathilda and Matt Share a Pie.... [01/20/2010]
Doctor Ian starts with simpler fractions, and a drawing, to model the parts, wholes, and parts-as-wholes in this problem of repeated division.

Means and Extremes [06/20/2001]
Why are certain parts of a proportion called the means and the other parts called the extremes?

Means and Extremes of a Proportion [09/16/2004]
In the proportion a/b = c/d, why are the a and the d called the extremes and the b and c called the means of the proportion?

M&M's and Percentages of their Colors [12/15/1995]
I have a question about M&M's: I have the old statistics about what percent of what color were made and am looking for the new stats with the new color blue.

Percent Greater Than vs. Increased [11/22/2002]
Is there a difference between the following statements? My profits are 200% greater than they were last year; My profits from last year have increased 200%.

Percent Increase and "Increase by a Factor of ..." [08/01/2006]
If we start with 1 apple today and tomorrow have 2 apples, then we have a 100 percent increase. But can't I also say there was an increase by a factor of 2?

Price of House, Commission [8/29/1996]
If a real estate agent recieves 6 percent of the selling price of each house she sells, then what should be the sale price of a house if a seller wants to keep $61,000?

Problems on Rates and Unit Conversions [06/21/1999]
Can you help me with problems on rates and unit conversions?

Proportionality Symbol [05/20/2003]
Do you know the name of the little symbol used in the topic of rates and variation that means 'in proportion to'?

Proportional Weights [08/21/1998]
Jim weighs 60 kilograms and needs to lift a 300-kilogram rock. His crowbar is 1.5 meters long. How close must the fulcrum be to the rock so that Jim's weight can lift the rock?

Proportion or Analogy? [12/12/2001]
What does 9:3::49: mean?

Proportions for a Chemical Solution [6/24/1996]
I need to mix up a 10 percent solution that contains a dry chemical...

Proportions of Exact Enlargements [03/18/1998]
How are two objects related if one is an "exact enlargement of the other"?

Proportion Solution Option Profusion [09/24/2014]
A teen's proportional reasoning cuts a swift path to solving ratio problems in a single variable. Doctor Peterson praises his initiative — then extends the student's insight.

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