PROPORTION PROBLEMS

WORD PROBLEMS (advanced)

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 ˝.

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 Advanced Proportion Problems

Some of the problems may be more easily solved by not using proportions but rather by using fundamental math operations such as addition, subtraction, multiplication and division. These problems can usually be identified by the amount of information given. They usually have more than 3 bits of information and the amount of information given is an even amount. To solve it, it is just a matter of reading the problem carefully and then thinking it through logically.

There may also be times where you will have to use some rudimentary math in conjunction with some proportion problem solving. These types of problems can usually be identified because the problem gives multiple comparisons and then asks you to solve for one of the comparisons. After simplification the amount of information to work with is of an odd amount.

Step 1 | Read the word problem carefully and completely. Determine what the word problem is asking you to find out.
Step 2 | Determine what information given in the word problem is needed to solve the problem, and what information is not needed to solve the problem.
Step 3 | Determine which mathematical and/or algebraic operation(s) will be needed to solve the word problem.
Step 4 | Set up the appropriate math operation(s) and simplify if necessary.
Step 5 | Form a proportion using two ratios. Use X for the unknown number.
Step 6 | Apply the Cross-product Rule to solve for X.

Example 1: The harbor was 3 rods wide and 10 rods long. Helga wanted to cut diagonally across the harbor in 11 minutes. Placido cut across a geometrically similar harbor with a diagonal of 14.996 rods. How long was Placido’s harbor? Round answer to the nearest hundredth.

Step 1 | Read the word problem carefully and completely. Determine what the word problem is asking you to find out.
The problem is asking us to find the length of Placido’s harbor.
Step 2 | Determine what information given in the word problem is needed to solve the problem, and what information is not needed to solve the problem.
We need to find the length of Placido’s harbor and we are given that his harbor is similar geometrically to Helga’s harbor. Time has nothing to do with the measurements, so the time given in the problem is just extra information which will not be needed. Eliminating the time and counting the remaining bits of information we can see that we have 3 bits of information with which to work with. Therefore, we will be solving this problem using proportions.
Step 3 | Determine which mathematical and/or algebraic operation(s) will be needed to solve the word problem.
We could probably understand this problem better if we draw it out. Helga’s Harbor:
Since we have the length and width of Helga’s Harbor and we are given the diagonal measurement of Placido’s Harbor, then we need to find Helga’s diagonal measurement so that we have something in common to both Harbors. We can do this by using Pythagorean’s Theorem. a2 + b2 = c2 32 + 52 = c2 9 + 100= c2 = c 10.440 = c Therefore, Helga’s diagonal measurement for her Harbor is 10.440 and Placido’s is 14.996. As you can see, Placido’s Harbor is smaller. So, let’s rough sketch them out and view them side by side: Placido’s Harbor: Helga’s Harbor:
Now we have all the information we need to solve this problem.
Step 4 | Set up the appropriate math operation(s) and simplify if necessary.
No extra steps are necessary
Step 5 | Form a proportion using two ratios. Use X for the unknown number.
Since we need to solve for the length of Placido’s Harbor, we will use the length of Helga’s Harbor (10) plus both of the diagonals (10.440 and 14.996.) in setting up our proportion.
Step 6 | Apply the Cross-product Rule to solve for X.
In the proportion, X is equal to 14.364.
14.364 rounded to the nearest hundredths is 14.36.
The length of Placido’s Harbor was 14.36 kilometers long.

Example 2: The industrialist took out a loan from Thumbs that paid interest of 235.0% every 8 months. He borrowed $2,690.74 for a total of 11 months. How much money did Tumbs expect from the industrialist? Round answer to the nearest cent.

Step 1 | Read the word problem carefully and completely. Determine what the word problem is asking you to find out.
The problem is asking us to find out how much money Tumbs is expecting to get back from the industrialist. This would be the interest owed plus the amount of money lent.
Step 2 | Determine what information given in the word problem is needed to solve the problem, and what information is not needed to solve the problem.
We need the percentage rate that Tumbs demands (235.0% for every 8 months) and the amount of the loan the industrialist took out ($2,690.74) and for how long he borrowed it (11 months). Counting the information that we need to solve this problem we see that we have 4 bits of information. Since this is an even amount we can solve this problem without using proportions.
Step 3 | Determine which mathematical and/or algebraic operation(s) will be needed to solve the word problem.
It appears that we will be using division because we need to find out how much interest the industrialist will be paying per month (235.0% divided by 8 months) and we will be using multiplication to multiply the interest per month by the 11 months which is the term of the loan. We will also be using addition to sum the interest due Tumbs plus the original load amount of $2,690.74.
Step 4 | Set up the appropriate math operation(s) and simplify if necessary.
First let’s calculate the amount of interest that the industrialist will pay Tumbs on a monthly basis: 235.0% ÷ 8 months = 29.375 % / month Convert this percentage to a decimal number by dividing by 100: 29.375 ÷ 100 = .29375 / month Next we need to figure out how much interest the industrialist will pay Tumbs for 21 months:
Multiplying this with the loan amount will give us the dollar amount of interest the industrialist paid Tumbs. While we are at it we might as well round up to the nearest cent. $2,690.74 x 3.23125 = $8694.453625 Lastly, add this interest paid to the original loan amount and we get the total amount paid to Tumbs. $2,690.74 + $8694.453625 = $11385.19. The total amount of monies paid to Tumbs was $11385.19
Step 5 | Form a proportion using two ratios. Use X for the unknown number.
Not a proportion problem
Step 6 | Apply the Cross-product Rule to solve for X.
Not a proportion problem
The total amount of money Tumbs expected from the industrialist was $11385.19.

Example 3: Polar bears were eating well since the iol platformas thawed. For every 6% of iol platformas that melted the Polar bear’s food supply increased by 7%. If the food supply increased by 12% what percent of the platformas melted? Round answer to the nearest hundredth.

Step 1 | Read the word problem carefully and completely. Determine what the word problem is asking you to find out.
The problem is asking us to find the percentage of platformas that melted.
Step 2 | Determine what information given in the word problem is needed to solve the problem, and what information is not needed to solve the problem.
It appears that all the information given will be useful in solving the problem. We can also see that we have 3 bits of information with which to work with. Therefore, we will be solving this problem using proportions.
Step 3 | Determine which mathematical and/or algebraic operation(s) will be needed to solve the word problem.
No extra steps are necessary
Step 4 | Set up the appropriate math operation(s) and simplify if necessary.
No extra steps are necessary
Step 5 | Form a proportion using two ratios. Use X for the unknown number.
Step 6 | Apply the Cross-product Rule to solve for X.
In the proportion, X is equal to 10.2857.
10.2857 rounded to the nearest hundredths is 10.29.
The percentage of platformas that melted was 10.29.

Example 4: Gullum wanted to buy Gullum Knapsacks for the long journey. At the Bag End Purveyors of finery they cost $205.68 per 19 Knapsacks plus tax of 10%. Gullum thought he would need 5 for the long journey, so how much would the total bill be? Round answer to the nearest cent.

Step 1 | Read the word problem carefully and completely. Determine what the word problem is asking you to find out.
The problem is asking us to find the total cost that Gullum would have to pay if he bought 26 Knapsacks.
Step 2 | Determine what information given in the word problem is needed to solve the problem, and what information is not needed to solve the problem.
At first glance it appears that we have 4 bits of information to work with but upon closer examination we can see that we can include the 10% into the cost of $205.68. This thereby reduces the count of information down to three. Therefore, we will be solving this problem using proportions.
Step 3 | Determine which mathematical and/or algebraic operation(s) will be needed to solve the word problem.
We need to include the tax of 10% into the price of 19 Knapsacks which is $205.68.
So, now we have our three pieces of information required to set up our proportion equation: 19 Knapsacks= $226.248 5 Knapsacks = x
Step 4 | Set up the appropriate math operation(s) and simplify if necessary.
No extra steps are necessary
Step 5 | Form a proportion using two ratios. Use X for the unknown number.
Step 6 | Apply the Cross-product Rule to solve for X.
In the proportion, X is equal to $59.538.
$59.538 rounded to the nearest cent is $59.54.
The total bill for 5 Knapsacks would be $59.54.

Example 5: For every 270 watermelon seeds that Mikiski plants she typically loses about 2% of them to the headless horseman and 8% burstinto flames upon contact with the soil. If she wants a yield of 380, how many seeds should she plant? Round answer to the nearest whole number.

Step 1 | Read the word problem carefully and completely. Determine what the word problem is asking you to find out.
The problem is asking us to find out how many spinach seeds Susanaki should plant in order to obtain a yield of 380 spinach plants.
Step 2 | Determine what information given in the word problem is needed to solve the problem, and what information is not needed to solve the problem.
It appears that all the information given will be useful in solving the problem. At first glance it appears that we have 4 bits of information to work with but upon closer examination we can see that we can combine Mikiski’s total loses by adding the 2% with the 8%. This thereby reduces the count of information down to three. Therefore, we will be solving this problem using proportions.
Step 3 | Determine which mathematical and/or algebraic operation(s) will be needed to solve the word problem.
1) We will need to combine her loses into a single figure. 2) We will need to figure out her current yield based on her loses 3) We will need to set up a proportion to figure out how many seeds she should plant based on her loses
Step 4 | Set up the appropriate math operation(s) and simplify if necessary.
1) First we will combine her loses into a single figure: 2) Next we figure out her current yield based on her loses: 3) Now we have all the information need to set up our proportion: 270 spinach seeds = a yield of 243 x spinach seeds = a yield of 380
Step 5 | Form a proportion using two ratios. Use X for the unknown number.
Step 6 | Apply the Cross-product Rule to solve for X.
In the proportion, X is equal to 422.222
422.222 rounded to the nearest whole number is 422.
The total amount of seeds that Mikiski should plant is 422.