PROPORTION 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 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 , 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: Neville bought 11 broomsticks for $33.97. If Neville had $189.57 to buy broomsticks, how much change would be left after buying as many as possible? Round answer to the nearest cent.

| By reading the problem we notice that we are given exactly 3 bits of information: (1) 11 wands, (2) $33 .97, and (3) $ 189.57 . Therefore, we will solve this problem using proportions. |
Step 1 | Form a proportion using two ratios. Use X for the unknown number. |
Step a | Form the first ratio using the dollar amounts given for the known and unknown wands. |
| We were given that the first dollar amount in the problem is $33.97 and that the second dollar amount is the $189.77. |
Step b | Form the second ratio using the data given for the total number of wands, both known and unknown. |
| Form a proportion using the two sets of data. And then rearrange the terms so that the unknown variable X is on top. |
Step 2 | Apply the Cross-product Rule to solve for X. (Which is the amount of wands that can be bought with $189.57) First, multiply 11 by $189.57. Then, divide $2085.27 by $33.97. So the amount of wands that can be bought is 61.386 Since we cannot buy a fraction of a wand then the nearest whole amount of wands that we can buy is 61. X equals 61. | 11 x $189.57 = $2085.27 $2085.27 ÷ $33.97 = 61.386 X = 61.386
| Since the problem is asking us, “how much change would be left after buying as many as possible” then we need to convert the fractional part of .386 back into dollars and cents. Gathering our data we have: .386 wands for $33.97 From our data we notice that once again we have 3 bits of information. Therefore, we will use proportions to solve for the dollar amount left over for the .386 wand. |
Step 3 | Form a proportion using two ratios. Use X for the unknown number. |
Step a | Form the first ratio using the amounts given for the known and unknown wands. |
| We were given that the first amount in the problem is 11 and that the second amount is the .386. |
Step b | Form the second ratio using the data given for the total cost of the wands, both known and unknown. |
| Form a proportion using the two sets of data. And then rearrange the terms so that the unknown variable X is on top. |
Step 4 | Apply the Cross-product Rule to solve for X. First, multiply .386 by $33.97. Then, divide $13.11 by 11. So the amount of change left over is 1.19 cents. X equals 1.19. | .386 x $33.97 = $13.11 $13.11 ÷ 11 = $1.19
Total amount of change left over is 1.19 cents.

Example 2: Igor has 6 quarters and 7 cents which could buy 14 purple polo pads. If Igor had 24 quarters and 26 cents, how many purple polo pads could Igor buy? Round answer to the nearest item.

| By reading the problem we notice that we are given exactly 3 bits of information: (1) 14 purple polo pads, (2) quarters and cents, and (3) quarters and cents. Therefore, we will solve this problem using proportions. |
Step 1 | Form a proportion using two ratios. Use X for the unknown number. |
Step a | Form the first ratio using the total amounts of money given in the problem. |
| We were given that the first amount in the problem is: 6 quarters plus 7 cents. (6 x 0.25 = $1.50 + 0.07 = $1.57) The second amount is 24 quarters and 26 cents. (24 x 0.25 = $6.00 + 0.26 = $6.26) |
Step b | Form the second ratio using the data given for the total amount of farmers that found gold, both known and unknown. |
| Form a proportion using the two sets of data. | =
Step 2 | Apply the Cross-product Rule to solve for X. First, multiply 6.26 by 14. Then, divide 87.64 by 1.57. Finally, round 55.82 to the nearest item 55. | 6.26 x 14 = 87.64 87.64 ÷ 1.57 = 55.82 X = 55.82
The number of purple polo pads Igor could buy would be 55.

Example 3: If 14 subs cost $246.04 and 28 onion rings cost $196.52, how many onion rings can be bought if you bought 37 subs and spent a total of $1,162.58? Round to the nearest item.

| 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 lets extract all of our information: (1) 14 subs cost (2) $246.04, and (3) 28 onion rings cost (4) $196.51 (they are made from a very rare onion). We are going to buy (5) 37 subs and we are going to spend a total of (6) $1,162.58. By looking at the above data we can see that we can divide (2) by (1) so that we will have just the cost for one sub. We can also do the same to get the cost for one onion ring, divide (4) by (3). That just leaves us with the total amount of money spent $1,162.58 and the amount of subs that will be bought –> 37. Therefore, 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 | Find the cost for each sub If 14 subs cost $246.04 then to determine the cost of just one sub we need to divide the $246.04 by 14. | $246.04 ÷ 14 = $17.574 1 sub = $17.574
Step b | Find the cost for each onion ring If 28 onion rings cost $196.51 then to determine the cost of just one onion ring we need to divide the $196.51 by 28. | $196.51 ÷ 28 = $7.018 1 onion ring = $7.018
Step c | Find the total amount of money spent for the 37 subs 37 subs at $17.574 a piece | 37 x $17.574 = $650.29
Step d | Find the total amount of money spent on onion rings The total amount of money spent for everything,$1,162.58, minus the cost of the subs $650.29 will give us the total amount of money spent on the onion rings. | $1,162.58 - $650.29 = $512.29
Step e | Find the total amount of onion rings bought Lastly, we take the total amount of money spent on the onion rings $650.29 and divide it by the cost of each onion ring $7.018 which will give us the total amount of onion rings bought. The number of onion rings bought is 73 | $512.29 ÷ $7.018 = 73
The amount of onion rings bought is 73

Example 4: Helga bought 7 nuclear subs for $234.97 with $7.69 change. How many can be bought with $215.65? Round to the nearest item that can be bought.

| At first glance it appears that we have 4 bits of information: (1) 22 nuclear subs, (2) $76.96 as the amount of money given for the subs, (3) $15.78 as the amount of change given back, and (4) $277.25 as the amount going to be spent for some nuclear subs. But if we look at it closely we can see that $15.78 was given back in change from the $76.96. By subtracting the $15.78 from the $76.96 then we would have the actual cost of the subs and we would also only have 3 bits of information. Therefore, we will solve this problem using basic math and proportions. |
Step 1 | Use basic math before forming the proportion |
| We need three bits of information to make a proportion. Therefore we will add the change of $15.78 back into the amount given of $76.96. Total amount of money spent = $76.96 - $15.78 Now we have the three bits of information to form our proportion. 1) 22 nuclear subs 2) $61.18 cost of the subs 3) $277.25 amount to be spent | $76.96 - $15.78 = $61.18
Step 2 | Form a proportion using two ratios. Use X for the unknown number. |
Step a | Form the first ratio using the total amounts of money given in the problem. |
Step b | Form the second ratio using the data given for the total amount of farmers that found gold, both known and unknown. |
| Form a proportion using the two sets of data. | =
Step 2 | Apply the Cross-product Rule to solve for X. First, multiply 277.25 by 22. Then, divide 6099.5 by 61.18. Finally, round 99.69 to the nearest item that can be bought, this would be 100. | 277.25 x 22 = 6099.5 6099.5 ÷ 61.18 = 99.69 X = 100
The number of subs that could be bought would be 100.