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Problem Solving
From Arithmetic to Algebra

Compare and Contrast:
Algebraic versus Arithmetic Solutions
In the following four problems, note how the arithmetic solutions use logic and involve a probing of the conditions of the problem—procedures which often easily transfer to the more abstract process of algebraic solutions, where problems ultimately are reduced to equations.

Problem #1
In a class with a total of 85 students, 380 oranges were distributed with each girl getting 5 oranges and each boy 4. How of the students are girls, and how many are boys?

Arithmetic Solution
380 oranges
85 students
5 oranges per girl
4 oranges per boy
Give each student 4 oranges: 4 x 85 = 340
380 – 340 = 40
There are 40 girls and 85 – 40 = 45 boys

Algebraic Solution (Incomplete)
380 oranges
85 students
5 oranges per girl
4 oranges per boy
Let x be the number of boys
Let y be the number of girls
4x + 5y = 380 and x + y = 85
Solve the above system to find x and y

Problem #2
A bottle and a cork together cost $1.10, but the bottle costs a dollar more than the cork. What is the cost of each?

Arithmetic Solution
Systematic Guess-and-Check Approach

Algebraic Solution (Incomplete)
Let b = cost of bottle
Let c = cost of cork
b + c = 1.10
b = c + 1.00
Solve the above system

Problem #3
One raja says to another, “If you give me one camel, then we will have an equal number of camels” The other replied, “No, if you give me one camel, then I will have double the number you have.” How many camels does each raja have?

Arithmetic Solution (Incomplete)
Teacher: By how much do the two numbers differ?
Student:
Teacher: Why?
Student:
(continued)

Arithmetic Solution (continued)
Teacher: Can one of the numbers be odd and the other, even?
Student: No
Teacher: Why?
Student:
(continued)

Arithmetic Solution (continued)
Teacher: Can both numbers be even?
Student: Yes
Teacher: Are you sure? What do we get, then, when 1 is added to the larger number? Is it even or odd? Could an odd number be the double of the lesser number?
(continued)

Arithmetic Solution (continued)
Student:
Teacher: So what do we know about the larger number? Is it even or odd?
Student: It’s odd.
Teacher: Why, then, must the smaller number be odd as well?
(continued)

Arithmetic Solution (continued)
Student:
Teacher: Can the numbers be large?
Student: No, when you add 1 to the larger number, it becomes double the smaller number.
Teacher: Let’s try some values.
(continued)

Arithmetic Solution (continued)

Algebraic Solution (Incomplete)
Let x be the number of camels owned by the first raja.
Let y be the number owned by the second.
x + 1 = y – 1
y + 1 = 2(x – 1)
Solve the above system.

Problem #4
Ninety-nine girls and one boy are in a mathematics class. How many girls must leave the room so the the percent of girls becomes 98 percent?

Arithmetic Solution (suggestion)
Use a guess-and-check strategy, focusing on the percent of boys in the class when the total number of students in the class is varied.

Algebraic Solution (incomplete)
Let x = the number of girls who must leave the class
99 - x   =  98
  100 – x      100
Solve the above proportion.

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