Question

Sadie sells half the paintings in her collection, gives one-third of her paintings to friends, and keeps the remaining paintings for herself. What fraction of her collection does Sadie keep?

Short answer

Sadie keeps 1/3 of her original collection of paintings.

Step-by-step solution

Determine fraction of paintings sold

Sadie sells half of the paintings in her collection. Therefore, the fraction of paintings sold is 1/2.

Determine fraction of paintings given away

Next, Sadie gives away one-third of the remaining paintings. After selling half, she has 1/2 left. Therefore, one-third of what's left would be \((1/3) * (1/2) = 1/6\). So, the fraction of paintings given away is 1/6.

Determine fraction of paintings kept

Finally, the remaining paintings are the ones that Sadie keeps. To find this, subtract the fractions of the paintings sold and given away from the total. In equation form, this is: \(1 - (1/2 + 1/6) = 1 - (3/6 + 1/6) = 1 - 4/6 = 2/6 = 1/3\). Therefore, Sadie keeps one-third of the original paintings.

Key concepts

Fractions

Understanding fractions is key to solving many problems in mathematics, especially when dealing with parts of a whole. A fraction is essentially a way to describe a division of a quantity into equal parts. The top number, known as the numerator, indicates how many parts you have. The bottom number, or the denominator, states how many equal parts the whole is divided into.

In Sadie's problem, we consider her collection of paintings as a whole. When Sadie sells half of her collection, she is left with the other half, which is another fraction of the complete set.

• Fraction sold: 1/2
• Fraction remaining after sale: 1/2

Next, when she gives away one-third of what's left, the process involves multiplication of fractions. To find one-third of one-half, multiply the fractions: \(\frac{1}{3} \times \frac{1}{2} = \frac{1}{6}\).

Mastering how to handle fractions by addition, subtraction, and multiplication is crucial for solving many mathematical problems effectively.

Algebra

Algebra is like a magical language for math. It lets you create equations and express mathematical ideas conveniently. In Sadie's problem, algebra helps to calculate how much of her collection she keeps. By setting up the problem as an equation, you clearly see how much of the total collection is left with Sadie.

First, acknowledge the total collection represented by 1 (or 100% of it). Then, subtract the sum of the sold paintings and those given away:\[ 1 - \left(\frac{1}{2} + \frac{1}{6}\right) = 1 - \frac{4}{6} = \frac{2}{6} = \frac{1}{3}\]

This algebraic way of representing the problem helps in understanding what happens to the collection step by step. It is simple yet effective, making the arithmetic accessible to anyone familiar with basic algebra.

Word Problems

Word problems are a fantastic way to apply math in real-life scenarios. They help strengthen analytical and critical thinking skills by placing numbers and equations into context. When approached correctly, they become fun puzzles that engage your problem-solving abilities.

In a word problem like Sadie's, the challenge is to translate words into numbers and operations. By breaking down the narrative:

• Selling half of the paintings means calculating \(\frac{1}{2}\) of the total.
• Next, giving away a third of what's left involves further fraction calculations.
• Finally, determine the fraction Sadie keeps by simplification.

Working through these steps reveals not only the answer but also deepens our understanding of how fractions and algebra interact. Patience and practice transform word problems from intimidating tasks to satisfying exercises.