How to Solve Rational Numbers Word Problems
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Indeed, tackling word problems that involve rational numbers may initially appear daunting. However, with a systematic methodology, a dose of endurance, and a touch of regular practice, you can conquer these numerical puzzles. Let's journey into the complex realm of rational numbers and unearth a thorough strategy to resolve word problems involving these kinds of numbers.
Mastering Word Problems Involving Rational Numbers: A Systematic Approach
Embark on the journey of resolving word problems involving rational numbers with this comprehensive guide:
Phase 1: Breaking Down the Problem
Start by carefully reading the word problem to grasp its core elements. Keep an eye out for key terms and values that indicate the nature of the problem. Document the rational numbers involved, highlighting them if necessary. Your understanding of the problem will steer the entire solving process.
Phase 2: Spotting the Unknowns
Next, pinpoint what the problem wants you to solve. This could be a certain value, a relationship, or possibly a set of values. Use symbols like 'x' or 'y' to denote these unknowns, making further calculations smoother.
Phase 3: Converting to Mathematical Language
Turn the word problem into a mathematical equation using the provided information. Interpret phrases like "is the same as" as equals (=), "added to" as plus (+), "minus" or "less" as subtract (-), "multiplied by" as times (x), "divided by" as division (÷). Note that rational numbers can take the form of fractions or decimals.
Phase 4: Crafting the Equation(s)
Drawing on the conversion, formulate the equation(s) required to solve the problem. Confirm that each equation accurately reflects the situations or conditions laid out in the problem. If the word problem involves ratios or proportions, your equation will probably incorporate fractions.
Phase 5: Solving the Equation(s)
Use your mathematical prowess to solve the created equations. The solution process might entail adding, subtracting, multiplying, or dividing both sides of the equation by the same (non-zero) number. Remember the bedrock rule of equations: Whatever you do to one side, you must do to the other.
Phase 6: Checking Your Solution
Plug your solution back into the original equations to verify them. If they stand true, you've successfully found your solution! If they don't, revisit your steps to find any mistakes in formulation or calculation.
Phase 7: Responding to the Question
Lastly, ensure to answer the question posed in the word problem. After all your hard mathematical work, it's about crafting your answer in a full sentence or as per the question's format.
Exercises for Rational Numbers Word Problems
1) Sophie spent \( \frac{1}{3} \) of her savings on a new phone and \( \frac{1}{4} \) on a pair of shoes. What fraction of her savings is left?
2) Michael ate \( \frac{3}{4} \) of a pizza and his friend, Ryan, ate \( \frac{2}{3} \) of another pizza of the same size. Who ate more pizza?
3) Jenny's recipe for lemonade requires \( \frac{1}{2} \) cup of sugar and \( \frac{1}{4} \) cup of lemon juice. How much more sugar does she need than lemon juice?
4) Aaron bought \( 2\frac{1}{2} \) yards of fabric to make shirts and used \( 1\frac{1}{3} \) yards for the first shirt. How much fabric is left?
5) A cake recipe calls for \( \frac{2}{3} \) of a cup of butter. If Liz only has \( \frac{1}{2} \) cup of butter, does she have enough for the recipe?
6) A car travels \( \frac{3}{5} \) of its journey. If the total journey is 400 miles, how many miles has the car traveled?
7) David read \( \frac{1}{2} \) of a book on the first day and \( \frac{1}{4} \) of the remainder on the second day. What fraction of the book does he still have to read?
8) A restaurant used \( \frac{4}{5} \) of its stock of vegetables one day. The next day, it used \( \frac{2}{3} \) of the remaining stock. What fraction of the initial stock of vegetables was left?
9) During a basketball match, James made \( \frac{5}{6} \) of his free throws while his teammate, Eric, made \( \frac{3}{4} \) of his. Who had a better free throw percentage?
10) Amanda and her friends baked \( \frac{2}{3} \) of a batch of cookies. They ate \( \frac{3}{8} \) of the baked cookies. What fraction of the total batch did they eat?
1) Solution
Sophie spent \( \frac{1}{3} + \frac{1}{4} = \frac{7}{12} \) of her savings. So, she has \( 1 - \frac{7}{12} = \frac{5}{12} \) of her savings left.
2) Solution
Michael ate \( \frac{3}{4} \) and Ryan ate \( \frac{2}{3} \). Converting to equivalent fractions with a common denominator, Michael ate \( \frac{9}{12} \) and Ryan ate \( \frac{8}{12} \). Therefore, Michael ate more pizza.
3) Solution
The recipe requires \( \frac{1}{2} - \frac{1}{4} = \frac{1}{4} \) cup more sugar than lemon juice.
4) Solution
Aaron has \( 2\frac{1}{2} - 1\frac{1}{3} = 1\frac{1}{6} \) yards of fabric left.
5) Solution
Liz does not have enough butter because \( \frac{1}{2} \) cup is less than \( \frac{2}{3} \) cup.
6) Solution
The car has traveled \( \frac{3}{5} \times 400 = 240 \) miles.
7) Solution
David read \( \frac{1}{2} + \frac{1}{2} \times \frac{1}{4} = \frac{1}{2} + \frac{1}{8} = \frac{5}{8} \) of the book. He still has \( 1 - \frac{5}{8} = \frac{3}{8} \) of the book to read.
8) Solution
The restaurant used \( \frac{4}{5} + \frac{1}{5} \times \frac{2}{3} = \frac{4}{5} + \frac{2}{15} = \frac{26}{30} = \frac{13}{15} \) of the vegetables. They were left with \( 1 - \frac{13}{15} = \frac{2}{15} \) of the initial stock.
9) Solution
James made \( \frac{5}{6} \) and Eric made \( \frac{3}{4} \). Converting to equivalent fractions with a common denominator, James made \( \frac{20}{24} \) and Eric made \( \frac{18}{24} \). Therefore, James had a better free throw percentage.
10) Solution
They baked \( \frac{2}{3} \) of the batch and ate \( \frac{3}{8} \) of the baked batch. They ate \( \frac{2}{3} \times \frac{3}{8} = \frac{1}{4} \) of the total batch.
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