Solving Math Problems: Book Pages and Least Common Multiple
Math problems can often be approached using various mathematical concepts and strategies. Two such problems presented here involve dealing with fractions and understanding the least common multiple (LCM). Let's delve into the detailed solutions for these problems.
Solving a Book Pages Problem Using Fractions
A student, Tarun, read three-eighths of a book one day, and then read four-fifths of the remaining pages the next day. After these two days, 30 pages were still unread. How many pages did the book contain?
To start, let's denote the total number of pages in the book as x.
On the first day, Tarun read 3/8 of the book, so he read:
[ frac{3}{8}x text{ pages} ]After the first day, the remaining pages are:
[ x - frac{3}{8}x frac{5}{8}x text{ pages} ]On the second day, Tarun read 4/5 of the remaining pages, which amounts to:
[ frac{4}{5} times frac{5}{8}x frac{4}{8}x frac{1}{2}x text{ pages} ]After the second day, the number of unread pages is given as 30. Therefore, the pages remaining after both days are:
[ frac{5}{8}x - frac{1}{2}x 30 ]Let's solve for x by finding a common denominator:
[ frac{5}{8}x - frac{4}{8}x 30 Rightarrow frac{1}{8}x 30 ]Multiplying both sides by 8 gives:
[ x 30 times 8 240 ]So, the book contains 240 pages.
To verify the solution, let's breakdown the pages read each day:
This matches the given information, confirming that the book indeed contains 240 pages.
Solving a Math Problem Using the Least Common Multiple
Another problem involves finding the least common multiple (LCM) and understanding its application. The least common multiple of 8 and 5 is 40. Let's solve this problem step-by-step.
Suppose the total number of pages is represented as 1. First day: ( frac{3}{8} ) of the book, which equals 15 units (since the LCM is 40, ( frac{3}{8} times 40 15 )) Remaining pages: ( 1 - frac{3}{8} frac{5}{8} ) of the book, equaling 25 units (since ( frac{5}{8} times 40 25 )) Second day: ( frac{4}{5} ) of the remaining (25 units), which equals 20 units (since ( frac{4}{5} times 25 20 )) Remaining pages: ( 25 - 20 5 ) units, and since 5 units of the total 40 units is given as 30 pages, the total number of pages in the book is ( 40 times 6 240 ) pages.
Understanding the Problem Using Different Strategies
Another approach involves using different fractions and the concept of reducing the fractions to simpler terms. First day: Tarun read ( frac{2}{9} ) of the book, leaving ( frac{7}{9} ) unread. Second day: Another ( frac{4}{5} ) of the remaining pages, calculated as ( frac{4}{5} times frac{7}{9} frac{28}{45} ). Remaining after the second day: ( frac{7}{9} - frac{28}{45} frac{35}{45} - frac{28}{45} frac{7}{45} ). Given that the ( frac{7}{45} 45 ) pages, the total number of pages in the book is ( 7 times 45 315 ) pages.
Conclusion
Each of these problems demonstrates how to approach complex mathematical problems using fractions and the least common multiple. These strategies are essential for solving a wide range of math problems, from basic arithmetic to more complex algebraic equations.
If you have similar math problems to solve, consider the following steps:
Define the variables clearly. Identify the fractions and apply the appropriate operations. Use the least common multiple to simplify fractions. Verify your solution by checking the given information.By practicing these steps, you can improve your understanding and proficiency in solving math problems.