4.3 Arithmetic Reasoning (Word Problems & Mathematical Logic)

Arithmetic Reasoning: Word Problems & Mathematical Logic

Arithmetic Reasoning on the OLSAT is not primarily a test of computational speed; it is a test of mathematical logic and modeling. The challenge lies in translating a paragraph of text into a workable mathematical equation or logical framework. You must decode the language of mathematics embedded within standard English prose.

Translating Words to Math

The first step in mastering Arithmetic Reasoning is learning the direct translations from English phrases to mathematical operations.

Key Translations:

• Equality ($=$): is, are, was, were, will be, yields, results in, equates to.
• Addition ($+$): sum, increased by, more than, combined, total, together.
• Subtraction ($-$): difference, decreased by, less than, fewer than, reduced by.
• Multiplication ($\times$): product, times, of (especially with fractions/percentages, e.g., "half of"), twice.
• Division ($\div$): quotient, divided by, per, ratio of, out of.

Example Translation: "Five less than twice a number is equal to the sum of the number and ten."

• Let the number be $x$.
• "Twice a number" $\rightarrow 2x$
• "Five less than" $\rightarrow - 5$ (Note: "less than" reverses the order, so it's $2x - 5$, not $5 - 2x$).
• "is equal to" $\rightarrow =$
• "the sum of the number and ten" $\rightarrow x + 10$
• Equation: $2x - 5 = x + 10$

Rate, Time, and Distance Problems

One of the most common categories is the Rate/Time/Distance problem, governed by the foundational formula:

Distance = Rate $\times$ Time ($D = R \times T$)

Strategies for Rate Problems:

1. Ensure Unit Consistency: If the rate is in miles per hour, but the time is given in minutes, you must convert the minutes to hours before multiplying.
2. Average Speed Trap: The average speed of a trip is NOT the average of the two speeds. It is the Total Distance divided by the Total Time. If you travel to a destination at 40 mph and return at 60 mph, your average speed is not 50 mph. You must calculate the total time taken for the entire journey to find the true average.

Work and Cooperative Task Problems

Work problems involve individuals or machines completing a task, often working together. The trick is to convert everything to a unit rate—how much of the task is completed in one unit of time.

Formula: If person A takes $a$ hours to complete a task, and person B takes $b$ hours, their combined rate working together is: $\frac{1}{a} + \frac{1}{b} = \frac{1}{t_{total}}$

Example: Alice can paint a room in 4 hours. Bob can paint the same room in 6 hours. How long will it take them working together?

• Alice's rate: $\frac{1}{4}$ of the room per hour.
• Bob's rate: $\frac{1}{6}$ of the room per hour.
• Combined rate: $\frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12}$ of the room per hour.
• To find the total time ($t$), solve $\frac{5}{12} = \frac{1}{t}$, which gives $t = \frac{12}{5} = 2.4$ hours.

Ratios and Proportions

Ratios compare two or more quantities. They can be expressed as "A to B," "A:B," or "$\frac{A}{B}$."

The 'Multiplier' Strategy

When given a ratio like 3:5 for red to blue marbles, think of the actual numbers as $3x$ and $5x$, where $x$ is a common multiplier. If you are told there are 40 marbles total, you can set up the equation: $3x + 5x = 40$ $8x = 40$ $x = 5$ Therefore, there are $3(5) = 15$ red marbles and $5(5) = 25$ blue marbles.

Basic Probability

Probability assesses the likelihood of an event occurring. It is calculated as:

Probability = (Number of Desired Outcomes) / (Total Number of Possible Outcomes)

Key Probability Concepts:

• Independent Events: The outcome of one event does not affect the other (e.g., flipping a coin twice). To find the probability of both happening, multiply their individual probabilities.
• Mutually Exclusive Events: Events that cannot happen at the same time. To find the probability of one OR the other happening, add their individual probabilities.

Mathematical Logic and Pattern Recognition

Some Arithmetic Reasoning questions are purely logical puzzles that use numbers. They might present a sequence or a matrix of numbers and ask you to find the missing value.

For these, look for:

• Arithmetic progressions (adding/subtracting a constant).
• Geometric progressions (multiplying/dividing by a constant).
• Interleaving patterns (e.g., the odd terms follow one rule, the even terms follow another).
• Relationships between columns and rows in matrices.

By systematically translating English into math, converting rates to unit values, and applying logical structures to puzzles, you can solve Arithmetic Reasoning questions accurately without complex calculations.