Question

In order to fulfill a local school's request for $x$ cakes, $B$ parents agreed to each bake an equal number of cakes. If $p$ of the parents did not bake any cakes, which of the following represents the additional number of cakes that each of the other parents had to bake in order for the school's request for $x$ cakes to be fulfilled? A. $$\frac{px}{B}$$ B. $$\frac{px}{B(B-p)}$$ C. $$\frac{x}{B-p}$$ D. $$\frac{p}{R_{m}r}$$ E. $$\frac{p}{B(B-p)}$$

Short answer

\frac{p x}{B(B - p)}

Step-by-step solution

Determine initial number of cakes per parent

Initially, there are B parents who were supposed to bake a total of x cakes. Each parent would have baked $\frac{x}{B}$ cakes.

Determine the number of remaining parents

Since p parents did not bake any cakes, there are now $B-p$ parents baking the cakes.

Total number of cakes needed to be baked by remaining parents

The total number of cakes to be baked remains x.!

Determine the new number of cakes each of the remaining parents needs to bake

Since $B-p$ parents are baking x cakes, each of these remaining parents needs to bake $\frac{x}{B-p}$ cakes.

Determine the additional number of cakes each remaining parent has to bake

Initially, each of the B parents was baking $\frac{x}{B}$ cakes. Now, each remaining parent has to bake $\frac{x}{B-p}$ cakes. The additional number of cakes each remaining parent has to bake is: $\frac{x}{B-p}-\frac{x}{B}$. This can be simplified to $\frac{px}{B(B-p)}$.

Key concepts

algebraic expressions

Understanding algebraic expressions is crucial for solving word problems, especially in exams like the GMAT. In this exercise, we deal with expressions to determine how many cakes each parent must bake. An algebraic expression represents numbers and variables along with arithmetic operations. Here, the expressions $\frac{x}{B}$ and $\frac{x}{B-p}$ show us how many cakes each parent would bake initially and after some parents didn't bake any cakes. By analyzing these expressions, we calculate the difference to find out the additional number of cakes each remaining parent needs to bake to meet the school's request.

division of labor

The division of labor concept helps in distributing tasks equally among all participants. It's an essential strategy in teamwork to ensure efficiency and productivity. In our exercise, the school requested a total of $x$ cakes, and initially, $B$ parents were to bake them. When $p$ parents didn't bake, the remaining $B-p$ parents divided the task among themselves. This led to each remaining parent having to bake more cakes than originally planned. The aim is to understand how the workload shifts when some team members can't contribute.

logical reasoning

Logical reasoning is about understanding and applying steps and rules to solve problems. It's especially useful in word problems like the one in the exercise. We start by finding how many cakes each parent initially needed to bake, then assess the situation after some parents can't bake. By using logical thinking, we determine the new number of cakes each remaining parent needs to bake. The logical steps lead us to find that the additional cakes each remaining parent needs to bake is represented by $\frac{px}{B(B-p)}$. This clear and systematic approach helps break down and solve complex problems.