One Easy Addition Model for a Billion Word Problems: Comparison Problems

Comparison problems are a sub-type of additive relationship problems common in school math and daily life, so it's good to recognize them quickly and apply the appropriate model to solve them.

Here's a brief review of the first part of this series found at http://tutorfi.com/Math/additionmodel:

The numbers 2, 3, and 5 are an addition "fact family."

Key related concepts for our addition model:

The sum of the two small numbers is the big number: 2 + 3 = 5

The difference between the big number and either of the small numbers (BIG - small) is the other small number:
5 - 3 = 2

The general model for additive relationship problems is:

Small#1 + small#2 = BIG#

Comparison problems involve two quantities being compared to each other and the difference between them. You'll often recognize them by "more" or "less" and "err" words: smaller, larger, heavier, lighter, taller, shorter, more expensive, etc. Comparison problems fit the additive relationship pattern: the small quantity and the difference add up to the BIG quantity.

The basic Comparison model is:

Small amount + difference = BIG amount

For example:
Peewee’s weight + weight difference = BRUNO’S weight


SOLVING COMPARISON PROBLEMS

When you are given any two of the numbers, or one of the numbers and a relationship between two of the numbers, you can find the missing number(s). The process is:

1. Identify the situation- You're considering the difference between two quantities being compared.

2. Following the generic model, use specific words to write a model for your problem.

Tip: Writing out the generic model first will help to etch the pattern deeply in your brain, especially if you say it as you write it! Follow the pattern of the generic model to write the specific model for your problem.

3. Summarize the information given and requested- This is very easy when given numbers, trickier when given relationships between numbers. Write the relationships with (semi-) algebraic expressions and test them with specific examples to make sure they're right. You may use variables in your expressions if you're comfortable doing this.

4. Replace the words of your model with the appropriate numbers and expressions.

5. Solve the resulting equation:
a. If you're finding the BIG amount, just add the two small numbers.
b. If you're finding the small amount or the difference, rearrange the equation and do a subtraction.
c. If there's a relationship involved, you'll need to do some algebraic-type work to find one number and go back to the relationship from Step 3 to find the other.

6. Check your answers to be sure they fit the information given and make sense. Redo if they don't.

7. When you're confident your solution is correct, write it in a complete sentence.



Here are two examples for solving comparison problems following the process.


Example 1
Given the difference and the BIG quantity, find the small quantity.

Your team beat your opponents by 35 points. Your team scored 87 points. How many points did your vanquished foe earn?

1. The model applies: two quantities (scores) are being compared and we're thinking about the difference between them.

2. Our specific model:
Losing score + difference = WINNING score

3. Given: WINNING score = 87, difference = 35. Find losing score.

4. Replace words with numbers in our model:
Losing score + difference = WINNING score becomes
Losing score + 35 = 87

5. Rearrange and subtract:
Losing score = 87 - 35
Losing score = 52

6. Check: 35 + 52 = 87, yes, and this makes sense.

7. Answer: Our foes earned a measly 52 points.

Are you thinking, "Good grief! I did this in my head in a second. This took forever!"? I know. But this is how you learn math skills - illustrate a powerful process on a simple problem to learn the process. Then apply the same process to more complicated problems. Doing this builds your algebraic thinking skills.


Example 2
Given a BIG quantity and a relationship between it and a small quantity, find the small quantity and the specific difference between the two quantities.

Bruno weighs 144 pounds. He weighs half again as much as (i.e. 50% more than) Peewee. How much does Peewee weigh, and what is the difference between their weights?

1. The model applies: We're comparing two weights and considering the difference between them.

2. Model:
Peewee's weight + weight difference = BRUNO’S weight

3. Given:
BRUNO'S weight = 144.
Relationship: Bruno weighs 50% more than Peewee. That means
Weight difference = .5 * Peewee's weight
Try it: if Peewee weighed 50 pounds, Bruno would weigh half that much more, i.e. 25 more pounds. Our expression works: 25 = .5 * 50.

4. Replace words with numbers and the relationship expression:
Peewee's weight + weight difference = BRUNO’S weight

Becomes

Peewee's weight + .5 * Peewee's weight = 144

5. Solve:
Peewee's weight + .5 * Peewee's weight = 144
1.5 * Peewee's weight = 144

Now divide both sides by 1.5:

(1.5 * Peewee's weight)/1.5 = 144/1.5
Peewee's weight = 96

Using the relationship in Step 3,
Weight difference = .5 * Peewee's weight = .5 * 96 = 48.

Another way to find the difference would be to subtract directly:
Weight difference = BRUNO'S weight - Peewee's weight = 144 - 96 = 48

6. Check: 96 + 48 = 144, and .5 * 96 = 48. True and these numbers fit the given information.

7. Answer: Peewee weighs 96 pounds, and Bruno weighs 48 pounds more than Peewee.


Remember! Practicing the model on simple problems will help you solve comparison problems and form good habits to use for trickier problems. This kind of algebraic thinking - model-based, linear and precise - is key to successful completion of a college degree and highly valued in the best-paying jobs.

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