Question

A convenience store has a "change bowl" on its counter in which there can be 5-cent nickels and/or 1 -cent pennies. The store manager insists that whenever there is a dollar (100 cents) or more in the bowl, some change must be removed. What expression gives range of $P$ pennies and $N$ nickels that could be in the change bowl at any given time without the cashier needing to remove any coins? (A) $100P-N>5$ (B) $6(P+N)<100$ (C) $0.01P+0.05N<100$ (D) $P+5N<100$

Short answer

The correct expression is option (D), $P+5N<100$.

Step-by-step solution

Understand the Problem

We need to determine the range of numbers of pennies ($P$) and nickels ($N$) that keep their total value under 100 cents. The total cents is given by the expression $P+5N$, because each nickel is worth 5 cents and each penny is worth 1 cent.

Formulate the Total Value Expression

Formulate the total value expression for the coins in terms of $P$ and $N$. The total value in cents of $P$ pennies and $N$ nickels is $P+5N$. The constraint is that this needs to be less than 100 to avoid removal.

Identify the Correct Inequality

Out of the given options, we need the inequality that corresponds to the constraint $P+5N<100$. Compare this with the choices provided: - Option (A) is $100P-N>5$, which is not consistent with our derived inequality.- Option (B) $6(P+N)<100$ does not match as well, as the coefficient of $P$ should be 1 and $N$ should be 5.- Option (C) translates to real values $0.01P+0.05N<100$, which is incorrect because it should equate to actual cents, not real-value interpretations.- Option (D) is $P+5N<100$, which matches our derived inequality exactly.

Key concepts

Inequalities in Mathematics

When dealing with inequalities in mathematics, we often encounter expressions that contain variables and compare the size or value of two or more expressions. An inequality is like an equation but instead of an equals sign, it includes inequality signs like <, >, ≤, or ≥. These symbols tell us if one side of an inequality is larger or smaller than the other.

In the context of our problem, we are working with the inequality $P+5N<100$, where $P$ stands for the number of pennies and $N$ for nickels. The inequality ensures that the total value of coins does not exceed 100 cents, which prevents the cashier from needing to remove some coins.

To solve inequalities, you'd often isolate the variable by performing operations similar to those you use with equations. However, be careful when multiplying or dividing by a negative number, as this action will reverse the inequality sign.

Coin Value Problems

Coin value problems are a classic type of math problem that requires you to understand the worth of different coins and how they contribute to a total amount. These types of problems are practical because they reflect real-life situations where we deal with money daily.

In our exercise, we have a specific scenario where the store wants to keep the sum of nickels and pennies below 100 cents. Each penny is worth 1 cent, and each nickel is worth 5 cents. So, for instance, if you have 10 pennies, you have 10 cents. If you have 10 nickels, you have 50 cents (i.e., $5\times 10$).

To express this in a mathematical way, you translate the coins into an equation. This kind of linear equation can be solved or manipulated using inequality rules to determine the possible combinations of coins that total less than the store’s threshold.

PSAT Test Preparation

Preparing for the PSAT involves familiarizing yourself with various math concepts, including inequalities and word problems like coin value issues. The PSAT tests your ability to apply mathematical reasoning in practical scenarios, which are very useful in your education and everyday life.

One effective way to prepare is to practice solving problems step by step, as demonstrated in the original step by step solution. Start by understanding what the problem is asking, set up your equations or inequalities, and then solve them systematically. Engage in regular practice with a variety of questions to build your confidence and efficiency.

Use resources such as practice tests, study guides, and tutoring sessions if needed. Tackling a wide range of problems will help sharpen your problem-solving skills and get you ready for test day. Remember, understanding the concept behind a problem is as crucial as finding the correct answer.