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Rational Number
Word Problems

Master real-world problems involving addition, subtraction, multiplication, and division of rational numbers, including negative quantities.

Pre-Algebra

What Are Rational Numbers?

A rational number is any number that can be written as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b \neq 0\). This includes integers, fractions, decimals that terminate, and decimals that repeat.

Integers
\(-5,\; 0,\; 12\) are all rational: \(\frac{-5}{1},\; \frac{0}{1},\; \frac{12}{1}\)
Fractions
\(\frac{3}{4},\; -\frac{7}{2},\; \frac{1}{8}\) are already in \(\frac{a}{b}\) form
Decimals
\(0.75 = \frac{3}{4}\) and \(0.\overline{3} = \frac{1}{3}\) are rational

Rational Numbers on the Number Line

\(-3\)
\(-2\)
\(-1\)
\(0\)
\(1\)
\(2\)
\(-2.5\)
\(-0.5\)
\(0.5\)
\(1.5\)
 Negative Positive

Why Word Problems Matter

Rational number operations appear everywhere in daily life. The challenge is translating English sentences into math expressions, then solving carefully with the correct signs.

Money
Deposits are positive, withdrawals are negative: \(+45.50\) and \(-20.75\)
Temperature
A drop of \(8.5^\circ\) from \(-3^\circ\) gives \(-3 + (-8.5) = -11.5^\circ\)
Elevation
Sea level is \(0\): a submarine at \(-150\frac{1}{2}\) ft is below

Core Sign Rules for Operations

1
Adding same signs: add the absolute values, keep the sign. \((-4) + (-7) = -11\)
2
Adding different signs: subtract the smaller absolute value from the larger, take the sign of the larger. \((-9) + 5 = -4\)
3
Subtracting: add the opposite. \(3 - (-8) = 3 + 8 = 11\)
4
Multiplying / Dividing: same signs → positive, different signs → negative.

Multiplication & Division Sign Chart

Signs Result
\( (+) \times (+) \)
\(+\) Positive
\( (-) \times (-) \)
\(+\) Positive
\( (+) \times (-) \text{ or } (-) \times (+) \)
\(-\) Negative

Gains vs. Losses: Choosing the Operation

Addition Context Subtraction Context
Keywords deposit, rise, gain, above, more withdraw, drop, loss, below, less
Sign meaning Positive: increase Negative: decrease
Example \(50 + 30 = 80\) \(50 + (-30) = 20\)
Key insight Both use addition: losses are added as negative numbers

The 4-Step Problem-Solving Strategy

1
Read and identify: Highlight the quantities and their signs. Words like "loss," "below," and "withdrew" signal negative values.
2
Translate to an expression: Write each quantity as a rational number and choose the correct operation (\(+\), \(-\), \(\times\), \(\div\)).
3
Compute with sign rules: Apply the sign rules from Slide 4. Work with fractions or decimals as given.
4
Interpret the answer: Translate the number back into context. A negative result means debt, below zero, or a loss.

Example 1: Bank Account

Maria has \($125.50\) in her account. She withdraws \($40.75\) and then deposits \($18.25\). What is her new balance?

Start
\(\$125.50\)
 Withdraw
\(-\$40.75\)
 Deposit
\(+\$18.25\)
\(125.50 + (-40.75) + 18.25 = \$103.00\)

Example 2: Temperature Change

At midnight the temperature was \(-6\frac{1}{2}°\text{F}\). By noon it rose \(15\frac{3}{4}°\text{F}\). What was the noon temperature?

\(-10\)
\(-5\)
\(0\)
\(5\)
\(10\)
Midnight: \(-6.5^\circ\text{F}\)
Noon: \(9.25^\circ\text{F}\)
\(+15.75^\circ\)
 Computation
\(-6.5 + 15.75\)
\( = 9.25^\circ\text{F}\)

Example 3: Scuba Diving Descent

A diver descends at a rate of \(2\frac{1}{3}\) feet per second. After \(6\) seconds, what is the diver's position relative to the surface?

Surface (\(0\) ft)
\(-5\) ft
\(-10\) ft
\(-15\) ft
\(-14\) ft
Rate \(\times\) Time
\( (-2\frac{1}{3})(6) = -14 \)
\(14\) ft below surface

Example 4: Splitting a Debt

Three friends owe a total of \($47.25\). They split the debt equally. How much does each person owe?

1
Identify: Total debt is \(-47.25\) (negative because it is money owed). Divide among \(3\) people.
2
Translate: \(\frac{-47.25}{3}\)
3
Compute: \(47.25 \div 3 = 15.75\). Since negative divided by positive is negative: \(-15.75\)
4
Interpret: Each person owes \($15.75\).

Example 5: Stock Price Changes

A stock changes by \(-\frac{3}{4}\) points on Monday, \(+1\frac{1}{2}\) on Tuesday, and \(-\frac{5}{8}\) on Wednesday. What is the total change?

Mon
\(-\frac{3}{4}\)
 Tue
\(+1\frac{1}{2}\)
 Wed
\(-\frac{5}{8}\)
Total: \(+\frac{1}{8}\) point

Common Mistakes to Avoid

✕ Dropping the Negative
Writing \(50 - 30 = 20\) when the problem says "lost \($30\)" from \($50\) in debt: the correct setup is \(-50 + (-30) = -80\).
✕ Wrong Sign on Product
Forgetting that \((-3) \times (-4) = +12\), not \(-12\). Two negatives multiplied yield a positive.
✕ Subtracting Instead of Adding Opposite
Writing \(5 - (-3) = 2\) instead of \(5 + 3 = 8\). Always convert subtraction of a negative to addition.

Operations Summary

Operation Rule Example
Add (same sign) Add, keep sign \(-4 + (-7) = -11\)
Add (diff. sign) Subtract, take larger's sign \(-9 + 5 = -4\)
Subtract Add the opposite \(3 - (-8) = 11\)
Multiply / Divide Same → \(+\), Diff → \(-\) \((-6) \div (-2) = 3\)
Mixed numbers Convert to improper/decimal first \(2\frac{1}{3} = \frac{7}{3}\)

Question 1 of 5

A submarine is at \(-120\) feet. It rises \(45\frac{1}{2}\) feet. What is its new depth?

Question 2 of 5

The temperature dropped \(3.5°\text{F}\) each hour for \(4\) hours. If the starting temperature was \(2°\text{F}\), what is the final temperature?

Question 3 of 5

Four friends share a restaurant bill. After a \($12\) coupon, the remaining bill is \($53.80\). Each person pays equally. What does each person pay?

Question 4 of 5

What is the value of \(-\frac{2}{3} \times \frac{9}{4}\)?

Question 5 of 5

A hiker starts at an elevation of \(-25\frac{1}{4}\) meters (below sea level) and climbs \(38\frac{3}{4}\) meters. What is the hiker's new elevation?