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Magazine Chapter 3: Problem 31
Find three consecutive numbers that have the sum of 117
Short Answer
Expert verified
The three consecutive numbers are 38, 39, and 40.
Step by step solution
01
- Define the consecutive numbers
Let's define the three consecutive numbers as follows: Let the first number be x, the second number will be x + 1, and the third number will be x + 2.
02
- Set up the equation
We know that the sum of these three consecutive numbers is 117. Therefore, we can set up the following equation: \[ x + (x + 1) + (x + 2) = 117 \]
03
- Simplify the equation
Combine like terms to simplify the equation: \[ x + x + 1 + x + 2 = 117 \] \[ 3x + 3 = 117 \]
04
- Solve for x
Subtract 3 from both sides of the equation: \[ 3x + 3 - 3 = 117 - 3 \] \[ 3x = 114 \]Next, divide both sides by 3: \[ x = \frac{114}{3} \] \[ x = 38 \]
05
- Identify the three consecutive numbers
With x = 38, the three consecutive numbers are: First number: x = 38 Second number: x + 1 = 39 Third number: x + 2 = 40
06
- Verify the solution
Check if the sum of these numbers is 117: \[ 38 + 39 + 40 = 117 \]. The sum is indeed 117, confirming our solution.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Problem-Solving Strategies
When tackling math problems, having a structured approach is crucial. For instance, in our consecutive numbers exercise, we use a methodical process to find the solution:
- First, define the unknown variables. Here, we set the consecutive numbers as x, x+1, and x+2.
- Then, set up an equation based on the given information. For this exercise, it was the sum equals 117.
- Next, simplify the equation by combining like terms. This helps us isolate the variable.
- Lastly, solve for the variable and verify the solution by plugging it back into the original context.
Basic Algebra
Our problem involves basic algebra to find three consecutive numbers. Algebra is a branch of mathematics that uses symbols to represent numbers and express mathematical relationships. Here are the key algebraic steps involved:
- Identifying variables: We assigned x as the first number, x+1 as the second, and x+2 as the third.
- Writing an equation: Using these variables, we formed the equation x + (x+1) + (x+2) = 117.
- Simplifying the expression: By combining the x terms, we got 3x + 3 = 117.
- Solving for the variable: Isolate x by performing the same operations on both sides of the equation. We subtracted 3 first, then divided by 3, resulting in x = 38.
GMAT Math Preparation
Preparing for the GMAT requires a strong foundation in various math concepts, including algebra. The consecutive numbers problem we just solved is a great example of the types of problems you might encounter on the test. Let鈥檚 see how this exercise aligns with GMAT math preparation:
- Variable manipulation: You need to be comfortable defining variables and forming equations, as we did by letting x be the first number.
- Equation setup: Crafting an equation from a word problem is a common GMAT task. We created x + (x+1) + (x+2) = 117 from the problem's description.
- Solving equations: Simplifying and solving for x requires basic algebra skills that are essential not just for the GMAT but for many real-world applications.
- Verification: Always double-checking your results ensures accuracy, which is crucial in high-stakes tests.
