Word Problems with Rational Numbers

Solving Word Problems with Rational Numbers

Rational numbers, consisting of fractions and integers, play a vital role in mathematical problem-solving. Word problems involving rational numbers are a common and valuable way to apply your understanding of these numbers in real-world contexts. In this article, we'll explore the various aspects of word problems related to addition, subtraction, division, and multiplication with rational numbers.

Addition with Rational Numbers

Addition problems with rational numbers typically involve combining like or unlike fractions. For example:

• Find the total number of $\frac{2}{3}$ cups of water if you have $\frac{1}{2}$ cup and $\frac{1}{3}$ cup of water.

To solve this, we'll add the numerators, while keeping the denominator the same:

[ \frac{1}{2} + \frac{1}{3} = \frac{1 \times 3 + 1 \times 2}{3 \times 1} = \frac{3}{3} + \frac{2}{3} = \frac{5}{3} \text{ cup of water.} ]

Subtraction with Rational Numbers

Subtraction problems with rational numbers often involve subtracting fractions from other fractions, integers from fractions, or fractions from integers. For instance:

• If you have $\frac{3}{4}$ cup of lemon juice and you pour $\frac{1}{2}$ cup, how much lemon juice is left?

To solve this, we'll subtract the numerators, while keeping the denominator the same:

[ \frac{3}{4} - \frac{1}{2} = \frac{3 \times 2 - 1 \times 4}{4 \times 1} = \frac{6 - 4}{4} = \frac{2}{4} = \frac{1}{2} \text{ cup of lemon juice.} ]

Division with Rational Numbers

Division problems with rational numbers often involve dividing fractions or integers by rational numbers, such as:

• If you have $\frac{2}{3}$ gallon of paint and you want to cover a $3 \times 1$ foot area, how much paint will you need per square foot?

To solve this, we'll divide the numerator by the denominator of the fraction:

[ \frac{2}{3} \text{ gallon per } 1 \text{ foot}^2 = \frac{2}{3} \text{ gallon} \div 1 \text{ foot}^2 ]

[ \frac{2}{3} \text{ gallon} \div 1 \text{ foot}^2 = \frac{2}{3} \text{ gallon} \times \frac{1 \text{ foot}^2}{1} = \frac{2 \text{ gallon}}{3 \text{ foot}^2} ]

[ = \frac{2}{3} \text{ gallon per } 3 \text{ foot}^2 = \frac{2}{9} \text{ gallon per } 1 \text{ foot}^2 ]

Multiplication with Rational Numbers

Multiplication problems with rational numbers typically involve multiplying fractions, integers, or mixed numbers by other fractions, integers, or mixed numbers. For example:

• If you have 2 apples and you buy $\frac{3}{4}$ pound of apples for $$1.50$, how much do the apples cost per pound?

To solve this, we'll multiply the price per pound by the amount of apples in pounds:

[ \frac{3}{4} \text{ pound} \times $1.50 = $0.75 \text{ per } \frac{1}{4} \text{ pound} ]

[ = $3 \text{ per } 1 \text{ pound} ]

Conclusion

Solving word problems with rational numbers requires patience and an understanding of the operations involved. Remember to follow a systematic approach, such as using the order of operations, to ensure that your calculations are accurate and that the solution makes sense in the context of the problem. Additionally, be sure to interpret the results in terms of the real-world context they represent.