RATIO PROBLEMS

SCALE

Concept: A ratio is a comparison of two numbers. It can be written in several ways. For example, when comparing the numbers 1 and 2, we can express the comparison as 1 to 2, or 1 : 2 or ½.

Always write the terms of the ratio in the same order as the problem compares them.

Just as we can reduce fractions to lowest terms, we can simplify ratios to their lowest terms, although a ratio with a denominator of 1 must still be expressed as a fraction, and ratios that appear to be improper fractions must be left as improper fractions.

Many ratio problems require more than one step.

You may not be given both of the numbers that you need to write a ratio. In this case, you may need to solve another equation to find the missing number.

Finally, when two ratios are written as equal ratios, the equation is called a proportion. For example, is a proportion. In many proportion problems, one of the four terms is missing.

The proportion can be solved using the Cross-product Rule.

Cross-product Rule
To find the missing number in a proportion, cross multiply, and divide the product by the third number. For example, in the proportion , first multiply . Then divide . In this problem .

Steps for Solving Simple Ratio Problems

Step 1 | Set up the ratio, writing the terms in the same order that the problem compares the terms.
Step 2 | Simplify the ratio to its lowest terms. (Remember that a ratio with a denominator of 1 must still be expressed as a fraction, and ratios that appear to be improper fractions must be left as improper fractions.)

Steps for Solving Multiple Step Ratio Problems

Step 1 | Set up the first ratio, writing the terms in the same order that the problem compares the terms.
Step 2 | Set up an equation to solve for the unknown number needed for the ratio.
Step 3 | Set up a ratio, writing the terms in the same order that the problem compares the terms.
Step 4 | Simplify the ratio to its lowest terms. (Remember that a ratio with a denominator of 1 must still be expressed as a fraction, and ratios that appear to be improper fractions must be left as improper fractions.)

Steps for Solving Simple Proportion Problems

Step 1 | Form a proportion using two ratios. Use X for the unknown number.
Step 2 | Apply the Cross-product Rule to solve for X.

Steps for Solving Multistep Proportion Problems

Step 1 | Set up the first ratio using given information.
Step 2 | Set up an equation, if necessary, to solve for any unknown information.
Step 3 | Set up another ratio, if necessary, using the new information.
Step 4 | Form a proportion using two ratios. Use the ratio from either Step 1 or Step 3 for the first ratio in the proportion, and another ratio, set up in the same order, while substituting X for the unknown number.
Step 5 | Apply the Cross-product Rule to solve for X.

Example1: The model destroyer is built on a scale of 1 inches to 5 feet. If the super structure of the model is 36 inches, the super structure of the actual destroyer is how many feet? Round the answer to the nearest hundredth.

Use the Steps for Solving Simple Proportion Problems

Step 1 | Form a proportion using two ratios. Use X for the unknown number.
We know that the model destroyer is built on a scale of 1 inches to 5 feet.
We want to find out how many feet the actual super structure is, if the super structure is only 36 inches on the model.
Step 2 | Apply the Cross-product Rule to solve for X.
To find the missing number in a proportion, cross multiply,
and divide the product by the third number.
In the proportion, X is equal to 180.
If a model destroyer is built on a scale of 1 inches to 5 feet, and the super structure of the model is 36 inches, then the super structure of the actual destroyer is 180 feet.

Example 2: While the Palace of Versailles was 82 yards high, the model of the Palace of Versailles was only 6.308 feet high. What is the scale of the model to the Palace of Versailles? Round the answer to whole numbers.

Use the Steps for Solving Simple Proportion Problems

Step 1 | Form a proportion using two ratios. Use X for the unknown number.
We know that the model Palace of Versailles is only 6.308 feet high. We know that the Palace of Versailles is 82 yards high.
We want to find out what the scale is of the model to the Palace of Versailles. First, match the units of measurement. (In this case, convert yards into feet.) becomes
Set up the proportion now, using only measurements in feet.
Step 2 | Apply the Cross-product Rule to solve for X.
To find the missing number in a proportion, cross multiply,
and divide the product by the third number.
In the proportion, X is equal to 38.998.
If the Palace of Versailles is 82 yards high, and the model of the Palace of Versailles is only 6.308 feet high, then the scale of the model to the Palace of Versailles is 1 to 38.998. Rounded to whole numbers, the scale is 1 to 39.

Example 3: He is carving a model ship from balsa on a scale of 1 : 17. If the actual ship is 28 yards long, the model aircraft carrier should be how many feet long? Round the answer

Use the Steps for Solving Simple Proportion Problems

Step 1 | Form a proportion using two ratios. Use X for the unknown number.
We know that the model ship is built on a scale of 1 to 17.
The actual ship is 28 yards long. We want to find out how many feet long the model is. (Note: Since the measurement we have been given is in yards, we will solve the problem in yards. We will convert the answer to feet at the end of the problem, since that is the unit of measurement the problem calls for the answer to be in.)
Step 2 | Apply the Cross-product Rule to solve for X.
To find the missing number in a proportion, cross multiply,
and divide the product by the third number.
In the proportion, X is equal to 1.64.
He is carving a model ship from balsa on a scale of 1 : 17. If the actual ship is 28 yards long, the model ship should be 1.64 yards long, or 4.92 feet long?
To determine how many feet are in 1.64 yards, set up a simple proportion and apply the Cross-product Rule. (Remember that there are three feet in one yard.)