See also the
Dr. Math FAQ:
3D and higher
Browse Middle School Ratio/Proportion
Stars indicate particularly interesting answers or
good places to begin browsing.
Selected answers to common questions:
Unit conversions, unit cancellation.
What is a ratio?
- Prorating the Rent [09/19/2002]
If my rent is $127 and I moved in on the 11th of this month, what
will my prorated amount be for September?
- Radius of a Sphere [01/29/1998]
What is the ratio of the areas and volumes of two spheres, one with
radius 3 times the other? What possible theorems are suggested?
- Raising the Antifreeze Ratio [08/07/99]
A 5-gallon radiator containing a mixture of water and antifreeze is
supposed to contain a 50% antifreeze solution...
- Ratio and proportion [12/5/1994]
I need extra help in ratio and proportion.
- Ratio and Proportion [7/4/1996]
Six men can complete a piece of work in one day... 44 men can build a
tower in 5 days...
- Ratio and Proportion: Beaches and Hawks [10/26/1998]
On a map, the scale states that 3 inches represent 125 miles. Two beaches
are 5.2 inches apart. How far apart are they in miles?
- Ratio and Proportion: Jobs [7/7/1996]
30 men can complete a job in 12 days at 8 hours/day. How many hours/ day
will it take to complete the job in 10 days with 24 men?
- Ratio of Anti-Freeze to Water [7/28/1996]
How much water should I add to a gallon of anti-freeze to make a 60/40
- Ratio of Apples to Pears to Oranges [09/13/2002]
If the ratio of apples to pears to oranges is 7:8:10, how many pears
are there if the total number of fruits is 500?
- Ratio of Greek to Latin Exams [06/10/2003]
Susanna's average score on all exams was 88. On Greek exams it was 80,
and on Latin exams 92. What is the ratio of the number of Greek exams
to the number of Latin exams?
- Ratio of Ore to Copper [9/4/1996]
Ore contains 2 percent copper. How much ore is needed to obtain 24 tons
- Ratio of Rates and Work [06/06/1998]
Given the rates of 5 semi-skilled and 4 skilled workers, can you find how
long it takes 1 semi-skilled and 2 skilled workers to do a job?
- Ratio of Ratios [06/05/2002]
In a community, 3/5 of the men are married to 2/3 of the women. What
fraction of the residents are married?
- Ratio of Sides and Ratio of Areas [02/11/1999]
If the sides of a triangle are in the ratio of 1:7, what is the ratio of
their areas? What about for other shapes?
- Ratio of Volume to Surface Area in Humans [12/09/1999]
What is the healthy ratio of volume to surface area for humans?
- The Ratio of Water to Wine [10/22/1995]
A cask is filled with 45 gallons of wine. Nine gallons are removed and
the cask is refilled with water. The nine gallons of the mixture are
removed and the cask is again refilled with water. What is the ratio of
water to wine in the final mixture?
- Ratios and Fractions: A Reconciliation [04/02/2013]
Teachers disagree about whether a fraction can compare part to part.
Doctor Peterson first distinguishes between ratios and fractions before reconciling them
and warning against inflexible thinking.
- Ratios and Geometry [10/29/1998]
An airplane flying at 33,000 feet has a visibility of 100 miles. What
percent of the total land area to the horizon is visible?
- Ratios and Mixtures [05/10/2001]
Sue needs to mix up a potion for her client... in what ratio must Sue mix
the honey and turpentine?
- Ratios and Proportions in Everyday Life [05/13/2002]
How are ratios and proportions used in everyday life?
- Ratios and Their Relative Benefits [03/04/2016]
Why use a ratio to compare numbers, rather than just subtracting them? By taking one example to extremes, and considering units, Doctor Peterson explains the advantages of proportional reasoning.
- Ratios Between Keys on a Piano [11/05/1997]
I know that there is a formula to describe a ratio from one key to
another but I can't find it...
- Ratios Containing a Zero [07/03/2005]
What does the ratio 1:0 mean? What does the ratio 0:1 mean? Are they
- Ratio Set-Up Problem [09/02/2003]
Concrete can be made by mixing cement, sand, and gravel in the ratio
3:6:8. How much gravel is needed to make 850 m^3 of cement?
- Ratios: Second Number [01/04/1998]
What does the second number in a ratio stand for?
- Recipes and Adjusting Ingredient Amounts [04/20/2005]
In a recipe, 5 1/2 cups of coconut uses 14 ounces of milk, but I only
have 4 cups of coconut, so how many ounces of milk do I use?
- Rule of Three [06/15/2002]
Please explain to me briefly (and with an example) the concept of the
rule of three.
- Scale Factor Acrobatics [07/15/2014]
A homeschooled teen struggles to make headway with a word problem in two
unknowns. After emphasizing the scale factor in their ratio, Doctor Ian models the
- Scale Factor of Similar Shapes [05/12/2000]
Find the scale factor, the ratio of the perimeters, and the ratio of the
areas of two regular octagons that have sides of lengths 21 and 28,
- Scaling up from a Numerator and Denominator to a Triple Ratio [07/08/2011]
A student familiar with fractions struggles to put more than two unknowns into one
proportional relationship. Doctor Jerry sets up equations to represent the scale factors.
- Setting up Proportions [03/11/2010]
Doctor Peterson explains how a proportion resembles a table of
data, and shows a seventh grader how to spot mistakes when
setting up relationships between ratios.
- Sides of Similar Triangles [06/11/1998]
The sides of a triangle are 24, 16, and 12. The shortest side of a
similar triangle is 6. Find the longest side of this triangle.
- Similar Rectangles [05/03/1997]
The outside boundary of an unfolded card is similar to its boundary when
it is folded. Find the width of the card if the open length is 8 and the
folded length is 4.
- Single Variable Algebraic Mixture Problem [05/03/2005]
Brittany needs a mixture of 76% oats and 24% corn. She currently has
25 pounds of a mixture that is 28% oats. How many pounds of oats
should be added to the mixture?
- Skilled and Semi-Skilled Workers [09/04/2002]
Four skilled workers do a job in 5 days, and five semi-skilled workers
do the same job in 6 days. How many days will it take for two skilled
and one semi-skilled worker to do that job?
- Solving a Ratio Problem [02/06/2006]
There are 33 red and yellow counters in a container. The ratio of red
to yellow is 3 to 8. How many of each color are in the container?
- Tea and Cakes [11/13/2002]
A cafe sold tea at 30 cents a cup and cakes at 50 cents each. Everyone
in a group had the same number of cups of tea and the same number of
cakes. The bill came to $13.30. How many cups of tea did each person
- Triangle and Trapezium Ratio [10/26/1998]
ABCD is a square with sides of length 4 cm. Find M on BC so that the
ratio of triangle ABM and trapezium ADCM is equal to 1/3.
- Turning a Perimeter into a Scale Factor [02/17/2003]
Perimeter and area ratios of similar figures are given. Find each
- Two Absolute Speeds versus a Single Relative One [08/06/2017]
A teen gets turned around modeling a question about two cars traveling the same
direction at different speeds. With insights into proportional reasoning and frames of
reference, Doctor Ian progressively steers her back to the path.