AI Tutor
Imagine a research submarine floats at 1121\frac{1}{2} miles below sea level (−112-1\frac{1}{2}), and then drops another 2342\frac{3}{4} miles. How deep is it now?
To find changes in depth, temperature, or money, we often need to subtract mixed numbers that have positive and negative signs.

Turn Subtraction into Addition

Subtracting any rational number is the exact same as adding its additive inverse (its opposite). That means a−b=a+(−b)a - b = a + (-b) and a−(−b)=a+ba - (-b) = a + b.
What is the safest way to compute with mixed numbers when signs get messy?

Strategy: Use Improper Fractions

An improper fraction is a fraction where the numerator is greater than or equal to the denominator, like 74\frac{7}{4}. Converting mixed numbers into improper fractions prevents regrouping errors when crossing zero.
Let's solve 114−3121\frac{1}{4} - 3\frac{1}{2}. First, rewrite it as addition: 114+(−312)1\frac{1}{4} + \left(-3\frac{1}{2}\right).
What happens when you subtract a negative mixed number from another negative mixed number?

Subtracting a Negative Across Zero

Let's calculate −213−(−456)-2\frac{1}{3} - \left(-4\frac{5}{6}\right). Since subtracting a negative is adding a positive, this becomes −213+456-2\frac{1}{3} + 4\frac{5}{6}.