Question

Solve Epigram 116: Mother, why do you pursue me with blows on account of the walnuts? Pretty girls divided them all among themselves. For Melission took two-sevenths of them from me, and Titane took the twelfth. Playful Astyoche and Philinna have the sixth and third. Thetis seized and carried off twenty, and Thisbe twelve, and look there at Glauce smiling sweetly with eleven in her hand. This one nut is all that is left to me. How many nuts were there originally? \(?^{22}\)

Short answer

Answer: There were originally 84 walnuts.

Step-by-step solution

Set up known fractions and quantities taken by each girl

We are given that Melission takes 2/7, Titane takes 1/12, Astyoche takes 1/6, Philinna takes 1/3, Thetis takes 20, Thisbe takes 12, and Glauce takes 11. The remaining walnuts for the child are 1. Write these quantities in terms of N: Melission: \(\frac{2}{7}N\) Titane: \(\frac{1}{12}N\) Astyoche: \(\frac{1}{6}N\) Philinna: \(\frac{1}{3}N\) Thetis: \(20\) Thisbe: \(12\) Glauce: \(11\) Child: \(1\)

Add the quantities and create an equation

Add all the walnuts taken by each girl and the child, which should be equal to the total number of walnuts N: \(\frac{2}{7}N + \frac{1}{12}N + \frac{1}{6}N + \frac{1}{3}N + 20 + 12 + 11 + 1 = N\)

Simplify the equation and solve for N

Combine the fractions and simplify the equation: \(\frac{41}{84}N + 44 = N\) Subtract \(\frac{41}{84}N\) from both sides of the equation: \(44 = \frac{43}{84}N\) Divide both sides by \(\frac{43}{84}\) to solve for N: \(N = \frac{84}{43} \times 44\) \(N = 84\) There were originally 84 walnuts.

Key concepts

Fractions

Fractions are a way to represent parts of a whole. They are written with two numbers: a numerator and a denominator. The numerator is the number of parts you have, while the denominator is the number of parts that make up a whole. Understanding fractions is crucial in solving problems like the walnut exercise where different portions are distributed among several people.

In this problem, each girl's share of walnuts is given as a fraction of the total number. For example, Melission takes \(\frac{2}{7}\) of the total walnuts, which means 2 out of every 7 walnuts were taken by her. When dealing with multiple fractions in a problem, it is important to find a common denominator to add or compare them easily.

A practical tip for working with fractions in word problems is to visualize them as parts of a pie or a set of items. This can help you understand and calculate the total more intuitively. By converting all taking fractions to a common base, you can add them together effectively to solve the equation and find the original total of walnuts.

Algebra

Algebra involves using symbols, typically letters, to represent numbers in equations. It is a powerful tool for solving problems where the exact quantities are not initially known. In the walnut problem, algebra helps us set up equations to find the original number of nuts before everyone took their share.

The first step in algebra is to define your variables. Here, \(N\) represents the total number of walnuts. The goal is to create an equation that shows all pieces of information given in the problem. Each girl's share is either a fraction of \(N\) or a specific number. By setting up an equation that adds all these parts, both fractions, and exact numbers, we can solve for \(N\).

Simplifying complex equations might involve combining like terms or fractions, as shown in the solution. Once the fractions are combined, subtracting and solving for the variable gives you the solution. Algebra is all about finding unknowns, and for this type of word problem, it gives a structured approach to organize information and derive answers logically.

Word Problems

Word problems are math questions presented in the format of a story or a real-life scenario. This makes them both relatable and sometimes tricky to solve, as they require converting words into mathematical equations. In the case of the walnut problem, we're given a narrative about several girls taking different parts of a total.

To tackle word problems efficiently, it's essential to read through the problem carefully and identify all the key details. Look for numbers, quantities, and relationships. For example, which fraction of walnuts each girl took, and how many specific nuts some of them seized. Each piece of information helps form the puzzle.

Creating a visual map or writing down each person's share can often help in organizing the information logically. Once organized, translate each piece into a mathematical expression or equation. Finding the balance between understanding the story and solving the math is what makes word problems unique.

Finally, checking your answer, as done in the problem step-by-step solution, ensures that all parts add up to the original number. This validation is crucial to ensure understanding and correctness, transforming word problems from daunting to manageable tasks.