Using Rates, Ratios, proportional Relationships and Scale Drawings

Rates, ratios, and proportional relationships all have something in common. They are comparisons of values with fractions. Carefully observe the differences between in the structure or the relationships shown below.

Watch the video below to examine how to find the missing hours given the miles and the miles/hour (pronounced miles per hour, meaning miles per 1 hour). Learn to use the formula in the video below. Watch out for the use of correct units!

Distance = rate times time or d = r * t or d = rt

Find d, r, or t depending on what's missing.

Solving ratios from word problems requires:

Associating the units in the fractional relationship described.
Changing the units of one to match the units of the other. Use the correct relationship ratio of the dimension needed/the dimension given, a form of one, ex. 12 inches / 1 foot.
Depending on the problem, lowest terms may not be appropriate.

Example 9

The rectangular garden is to be 6 yards long by 15 feet wide. Write the ratio of length to width.
$LaTeX: \frac{6\:yards}{15\:feet}=\frac{6\:yards}{15\:feet}\cdot\frac{3\:feet}{1\:yard}=\frac{18\:feet}{15\:feet}$
The garden will be 18 feet long and 15 feet wide. Note that reducing to lowest terms would not give the length and width of the garden so this answer should not be reduced.

Note: by means times

Solve Proportional Problems using these steps:

Write two fractions with an = between them.
Keep like terms in the same location of the proportion (both in the numerators or both in the denominators).
Use a variable to hold the place of the missing item.
Cross multiply.
Solve for the variable.

Example 10

Alyssa wants to create a smaller version of her neighbors garden which is 20 feet by 8 feet. She wants the longest side of her garden to be 12 feet. What is the length of the smaller side?

$LaTeX: \frac{20}{8}=\frac{12}{x}$
20x = 96
x = 4.8 feet
For a ruler .8 ft * 12 in/ft = 9.6 inches

Scale Drawings or Models

Scale drawings or model represent the real object in a different scale. A picture of a dog in a book is a smaller drawing, keeping the dimensions of the dog two dimensionally the same so that the dog is visually the same, just smaller.

An architectural company wishing to show a customer a replica of the new building design creates a scale model, a smaller version of the building which has all dimensions proportionally smaller than the new building will be when built. Models are three dimensional.

Indication measurement dimensions for scale drawings is normally represented with a colon, : , rather than a division sign , /. The scale of the dog to the picture is 10:1. This means that the dog is 10 times as big as the picture drawn. So if the length of the dog's body in the picture is 3 inches, then the dog's real length is 30 inches.

$LaTeX: \frac{10}{1}=\frac{x}{3}$

x = 30

Example 11

Note the picture of the squares below. The scale of the smaller square to the larger square is 1:3. The smaller square was used to create the larger so that the dimensions are easily seen.

image of one green square on left and nine green squares forming a square on the right

If the dimension of the larger square is 21 inches by 21 inches, what is the dimensions of the smaller square?

Note: "by" means times.

$LaTeX: \frac{1}{3}=\frac{x}{21}$

3x = 21

x = 7 inches

The smaller square is 7 inches by 7 inches.

The dimensions may also be written without the word by. Many times the dimensions are written with the letter x between them. The x in the case of a dimensional area means the word by, 7 inches x 7 inches and does also stand for times as the area of the square would be 49 square inches.

Note that solving scale drawing problems uses the same method of cross multiplication as the proportions, as they indeed are proportions of scale.

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