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Chapter 21: Problem 5

Three positive numbers \(x, y\), and \(z\) have the following relationships \(y=x+2\) and \(z=y+2\). When the median of \(x, y\), and \(z\) is subtracted from the product of the smallest number and the median, the result is 0 . What is the value of the largest number? (A) \(-2\) (B) \(\pi\) (C) 5 (D) 8 (E) \(21 / 2\)

Short Answer

Expert verified
The largest number is 5.

Step by step solution

01

Identify the relationships

Based on the problem, we know that:- \( y = x + 2 \)- \( z = y + 2 \). Therefore, \( z = (x + 2) + 2 = x + 4 \).
02

Determine the order of x, y, and z

Since \( z = x + 4 \), \( y = x + 2 \), and \( x \) itself, they are ordered as follows:\( x \), \( y = x + 2 \), and \( z = x + 4 \).
03

Identify the median

For the numbers \( x \), \( y = x + 2 \), and \( z = x + 4 \), the median is \( y \) since it is the middle value when they are arranged in ascending order.
04

Write the equation based on conditions

We are told that when the median is subtracted from the product of the smallest number (\( x \)) and the median (\( x + 2 \)), the result is 0. So the equation is:\[ x(x + 2) - (x + 2) = 0 \]
05

Solve the equation

Factor \( x(x + 2) - (x + 2) \) to get:\[ (x + 2)(x - 1) = 0 \]Solving for \( x \), we get \( x + 2 = 0 \) or \( x - 1 = 0 \).Since \( x \) is positive, \( x = 1 \).
06

Calculate y and z

Substitute \( x = 1 \) into the relationships:- \( y = x + 2 = 1 + 2 = 3 \)- \( z = x + 4 = 1 + 4 = 5 \)
07

Identify the largest number

Among \( x = 1 \), \( y = 3 \), and \( z = 5 \), the largest number is \( z = 5 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equations
Linear equations form the building blocks of algebra and are an essential concept to master in GMAT math problem solving. They are equations that graph as straight lines when plotted on a coordinate plane. In their simplest form, they appear as \( y = mx + b \), where:
  • \(y\) is the dependent variable,
  • \(m\) is the slope of the line,
  • \(x\) is the independent variable, and
  • \(b\) is the y-intercept.
For this particular exercise, we use linear equations to define relationships between the numbers \(x, y,\) and \(z\). Understanding that \(y = x + 2\) and \(z = y + 2\) helps us find expressions for each variable in terms of \(x\). This, in turn, allows us to solve for their values systematically, highlighting the power of linear equations in solving algebraic expressions.
Median Calculation
Calculating the median is a crucial step in various math problems. The median of a set of numbers is the middle value when they are ordered from smallest to largest. If the number of values is odd, the median is simply the middle number. If it's even, the median is the average of the two middle numbers.
In this exercise, the numbers are \(x, y = x + 2,\) and \(z = x + 4\). These values are naturally ordered, with \(x\) being the smallest, \(y\) the median, and \(z\) the largest. Therefore, the median here is \(y = x + 2\). Recognizing this sequence aids in forming equations and logic statements to solve for the value of \(x\).
Algebraic Expressions
Algebraic expressions are mathematical phrases involving numbers, variables, and operation symbols. They are pivotal in translating word problems into solvable mathematical equations. In this exercise, establishing the expressions for \(y\) and \(z\) in terms of \(x\) simplifies solving the problem:
  • \(y = x + 2\)
  • \(z = x + 4\)
Additionally, the problem's condition provided us with an expression to work with: \(x(x + 2) - (x + 2) = 0\). This equation is essential for finding \(x\), as we factor it to reveal that \(x = 1\). Understanding how algebraic expressions can be created and manipulated is central to excelling in GMAT math problem solving.
Problem-Solving Strategies
Mastering problem-solving strategies is key to dominating GMAT math exercises and similar high-pressure exams. Here, we combined logical thinking, sequence arrangement, and algebraic manipulation to find our answer. An effective approach involved:
  • Identifying relationships between variables.
  • Ordering numbers to quickly establish the median.
  • Setting up the required equation based on given conditions.
Solving the provided equation and substituting back to find all variables demonstrates a process of piecing together information methodically. Developing such strategies will equip you to tackle even complex problems with confidence and accuracy.

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Most popular questions from this chapter

If \(x

The median is larger than the average for which one of the following sets of integers? (A) \(\\{8,9,10,11,12\\}\) (B) \(\\{8,9,10,11,13\\}\) (C) \(\\{8,10,10,10,12\\}\) (D) \(\\{10,10,10,10,10\\}\) (E) \(\\{7,9,10,11,12\\}\)

Sarah cannot completely remember her four-digit ATM pin number. She does remember the first two digits, and she knows that each of the last two digits is greater than 5. The ATM will allow her three tries before it blocks further access. If she randomly guesses the last two digits, what is the probability that she will get access to her account? (A) \(1 / 2\) (B) \(1 / 4\) (C) \(3 / 16\) (D) \(3 / 18\) (E) \(1 / 32\)

A housing subdivision contains only two types of homes: ranch-style homes and townhomes. There are twice as many townhomes as ranch-style homes. There are 3 times as many townhomes with pools than without pools. What is the probability that a home selected at random from the subdivision will be a townhome with a pool? (A) \(1 / 6\) (B) \(1 / 5\) (C) \(1 / 4\) (D) \(1 / 3\) (E) \(1 / 2\)

A hat contains 15 marbles, and each marble is numbered with one and only one of the numbers 1, 2, 3. From a group of 15 people, each person selects exactly 1 marble from the hat. $$ \begin{array}{cc} \text { Numbered Marble } & \begin{array}{c} \text { Number of People Who } \\ \text { Selected The Marble } \end{array} \\ 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{array} $$ What is the probability that a person selected at random picked a marble numbered 2 or greater? (A) \(5 / 15\) (B) \(9 / 15\) (C) \(10 / 15\) (D) \(11 / 15\) (E) 1
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