RATIO PROBLEMS

MONEY (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 , and then divide . In this problem .

Steps for Solving Money 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.

For those problems that require solving by proportions the following steps should be followed:

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.

Example 1: Jeremy’s recipe called for 4 lbs of pork loin ($57 per pound), 7 cans of brown gravy ($0.65 per can), and 10 packages of spaghetti ($1.29 each) to feed 9. What is the total cost to feed 36? Round answer to the nearest hundredth.

| By reading the problem we notice that we are given 7 bits of information: (1) 4 pounds of pork loin at a price of (2) $57 per pound, (3) 7 cans of brown gravy at a price of (4) $0.65 per can, (5) 10 packages of spaghetti at a price of (6) $1.29 each, and that all that food will feed a total of (7) 9 people. If we combine the total cost of all the items, include the people that it feeds and include the amount of people we want to feed then we would only have 3 items that we need to work with. Therefore, we will solve this problem using proportions. |
Step 1 | Form a proportion using two ratios. Use X for the unknown number. |
Step a | Before we form our proportion we need to combine our dollar amounts. Total cost for 4 pounds of pork loin Total cost for 7 cans of brown gravy Total cost for 10 packages of spaghetti Total amount for all food items: | 4 x 57.00 = 228.00 7 x 00.65 = 004.55 10 x 01.29 = 013.60 -------- ----------------- 0 x 00.00 = 245.45
Step b | Form the ratio using the data given for the total amount of money it costs to feed 9 and 36. |
Step 2 | Apply the Cross-product Rule to solve for X. First, multiply 245.45 by 36. Then, divide 8836.2 by 9. Finally, round 981.8 to the nearest hundredth 981.8. | 245.45 x 36 = 8836.2 8836.2 ÷ 9 = 981.8 X = 981.8
To feed a total of 36 it would cost $981.8.

Example 2: In the catalog, gerbils were $42 each and mice cost only $5 each. If Dewlap bought a total of 153 animals and paid a total of $6,130.00, how many mice did he buy? Round answer to the nearest whole number.

| By reading the problem we notice that we are given more than 3 bits of information. Therefore we need to determine how best to solve this problem. Do we use basic math or do we use basic math with proportions? First let’s extract all of our information: (1) gerbils cost $42 each, (2) mice cost $5 each, (3) a total of 153 animals were bought, and (4) a total amount of $6,130.00 was spent. We have 4 bits of information with which to work with. This is an even amount therefore this problem can be solved using basic math without proportions. |
Steps | Use basic math to solve the problem |
Step a | The first equation that we can create is the sum of all the animals. G = gerbils and M = mice G + M = 153 | G + M = 153
Step b | Part of the second equation would include the total price paid for all the gerbils. Since each gerbil costs $42 then G x $42 = $42G | $42G
Step c | The other part of the second equation would include the total price paid for all the mice. Since each mice costs $5 then M x $5 = $5M | $5M
Step d | Now we can create the second equation that would be the total amount of money spent on all the animals. This is done by summing Step b and Step c. | 42G + 5M = 6130
Step e | Solve equations in Step a and in Step d: We want to solve for the number of gerbils bought. Therefore, we will rewrite the equation from Step a and put it in terms of Gerbils (G). From Step d, plug G into the equation. Multiply the 5 with what is in the parenthesis. Combine like terms. Solve for G The number of mice bought is 8 | G + M = 153 G = 153 – M 42G + 5M = 6130 42G + 5(153 – G) = 6130 42G + 765 -5G = 6130 37G = 6130 – 765 37G = 5365 G=5365 ÷ 37 G=145 M=M –G M=153 – 145 = 8
The amount of mice bought is 8

Example 3: At the haberdashery, strawberry cummerbunds were priced 1:1 of the lilac cummerbunds. Barney bought both colors and spent a total of $3,600.00. If Berto bought lilac cummerbunds in the ratio of 5 stawberry to 1 lilac, how much did Berto spend on lilac cummerbunds? Round answer to the nearest whole number.

| This problem is already giving us proportions to work with. So it is now a matter of organizing our data and thinking through this problem carefully. We are given: (1) stawberry to lilac cummerbunds are priced at 1:1, (2) $3600.00 as the total amount spent, (3) leggings bought in ratio of 5:1 – stawberry to lilac. Therefore, we will solve this problem using proportions. |
Step 1 | Form a proportion using two ratios. Use X for the unknown number. |
Step a | Since the proportions are already given to us, we do not need to form proportions but rather we need to organize our data and solve the problem directly. Ratio price of leggings: We’ll use S for stawberry and L for lilac. Ratio of amount bought (5 stawberrys to every 1 lilacs): | 5S = 1L
Step b | Multiply the two ratios to derive the cost-item ratio: We know that Barney bought 5 strawberry cummerbunds for every 1 lilacs leggings. Therefore, we will make it into a ratio. We will now bring down the cost ratio which tells us how much each item costs in a ratio format. By multiplying the 5 stawberrys that Barney bought with the cost ratio, and by multiplying the 1 lilacs that he bought by the cost ratio, we get the total amount of each leggings that he bought. Now we have the cost-item for each product: Stawberry = 5 Lilac = 1 | 5S = 1L
Step c | How many of each did he buy? He bought some quantity x times the cost-item of each product. In equation form: x(Stawberry + Lilac)=$3,600.00 Solve for x: | x(S + L)=3600 x(5 + 1) = 3600 x(6)= 3600 x=3600 / 6 x=600
Step d | Solve for the number of lilac cummerbunds bought. x(Stawberry + Lilac)=$3600.00 From the above equation we take out the Lilac portion to solve for the number of Lilac cummerbunds bought: x(Lilac) The number of Lilac cummerbunds bought was 600 | x(S + L)=3600 x(L) = 600(1) x(L) = 600
The number of lilac cummerbunds that were bought was 600.