The flowchart represents a mathematical algorithm that takes two positive integers as the input and returns a positive integer as...
GMAT Graphics Interpretation : (GI) Questions
The flowchart represents a mathematical algorithm that takes two positive integers as the input and returns a positive integer as the output. Processes are indicated in the rectangular symbols in the flowchart. Each process is symbolized by an equation, such as \(\mathrm{T = T + a}\). In this particular process, the current values of the variables T and a are added together and the sum then becomes the value of T. For example, if the value of T is 3 and the value of a is 7 before the process \(\mathrm{T = T + a}\) is completed, then the value of T is 10 and the value of a is 7 after the process is completed.
Use the drop-down menus to fill in the blanks in the following statements based on the algorithm represented by the flowchart.
Owning the Dataset
Table 1: Text Analysis
| Text Component | Literal Content | Simple Interpretation |
|---|---|---|
| Algorithm Type | "mathematical algorithm" | It's a specific procedure for solving a math problem |
| Input Requirement | "takes two positive integers as the input" | The algorithm requires two whole numbers greater than zero, named a and b |
| Output Type | "returns a positive integer as the output" | The result will be one whole number greater than zero |
| Process Symbols | "Processes are indicated in the rectangular symbols" | Calculation steps are shown in rectangles in the diagram |
| Equation Example | "such as \(\mathrm{T = T + a}\)" | This box means add 'a' to 'T' and update T with the result |
| Variable Explanation | "current values of the variables T and a are added together and the sum then becomes the value of T" | '\(\mathrm{T = T + a}\)' means add a to T and set T to the new value |
| Calculation Example | "if the value of T is 3 and the value of a is 7 before the process \(\mathrm{T = T + a}\) is completed, then the value of T is 10 and the value of a is 7 after" | After executing \(\mathrm{T = T + a}\), T becomes 10 (3+7) and a stays 7 |
Table 2: Chart Analysis
| Chart Component | What's Shown | What This Means |
|---|---|---|
| Start/End | Start and Stop points | Marks the beginning and end of the process |
| Initialization | \(\mathrm{T = 0}\) | The result variable T always starts at 0 |
| Input Step | Enter a and b | User provides two numbers for the algorithm |
| Decision: Is b odd? | Branches for yes/no | The algorithm checks if b is odd to decide the next step |
| If Yes (b odd) | \(\mathrm{T = T + a; b = b - 1}\) | Adds a to T and decreases b by 1, making it even |
| If No (b even) | \(\mathrm{a = 2a; b = b / 2}\) | Doubles a and halves b when b is even |
| Looping | Returns to check Is b odd? unless \(\mathrm{b = 1}\) | Process repeats steps until b reaches 1 |
| Terminal Decision | Is \(\mathrm{b = 1}\)? Then final steps | When b is 1, adds a to T, outputs T, then stops |
Key Insights
The algorithm uses a loop where b is reduced each time until it reaches 1. If b is odd, a is added to T and b is decremented. If b is even, a is doubled and b is halved. This is similar to Russian Peasant Multiplication, efficiently calculating \(\mathrm{a × b}\) using addition, doubling, and halving. T accumulates the correct sum by only adding a when b is odd, and the process always finishes since b steadily decreases.
Step-by-Step Solution
Question 1: Finding the First Nonzero Value of T in the Algorithm
Complete Statement:
If 24 and 35 are entered as the values for a and b, respectively, then the first nonzero value of T is ______.
Breaking Down the Statement
- Statement Breakdown 1:
- Key Phrase: 24 and 35 are entered as the values for a and b
- Meaning: We start the process by setting \(\mathrm{a = 24}\) and \(\mathrm{b = 35}\).
- Relation to Chart: These values are put into the flowchart at the 'Enter a and b' step.
- Important Implications: These specific numbers will control how the loop and decisions operate.
- Statement Breakdown 2:
- Key Phrase: first nonzero value of T
- Meaning: We are to find the first value T takes on that is not zero, as the algorithm runs.
- Relation to Chart: T starts at 0 and is only changed after certain operations depending on b's value.
- Important Implications: We are looking for the exact value produced when T first is updated from its initial value.
What is needed: The value that T first takes when it becomes not zero after beginning the process with \(\mathrm{a = 24}\) and \(\mathrm{b = 35}\).
Solution:
- Condensed Solution Implementation:
Trace the first step of the algorithm with \(\mathrm{a = 24}\) and \(\mathrm{b = 35}\), focusing on how and when T is changed from 0. - Necessary Data points:
Initial values: \(\mathrm{a = 24, b = 35, T = 0}\). - Calculations Estimations:
Since \(\mathrm{b = 35}\) is odd, the algorithm immediately computes \(\mathrm{T = T + a}\), so \(\mathrm{T = 0 + 24 = 24}\). This is T's first nonzero value. - Comparison to Answer Choices:
The answer choices likely include 24. This matches our result.
FINAL ANSWER Blank 1: 24
Question 2: Value of b After the Second Halving
Complete Statement:
If 35 and 27 are entered as the values for a and b, respectively, then after the process \(\mathrm{b = b/2}\) is completed for the second time, the value of b is ______.
Breaking Down the Statement
- Statement Breakdown 1:
- Key Phrase: 35 and 27 are entered as the values for a and b
- Meaning: We start by setting \(\mathrm{a = 35}\) and \(\mathrm{b = 27}\).
- Relation to Chart: These are the starting values input into the algorithm.
- Statement Breakdown 2:
- Key Phrase: after the process \(\mathrm{b = b/2}\) is completed for the second time
- Meaning: We're to keep track of each execution of \(\mathrm{b = b/2}\) and determine the value after the 2nd time it happens.
- Relation to Chart: \(\mathrm{b = b/2}\) occurs only when b is even, so we need to trace through the relevant steps and count these occurrences.
What is needed: The value of b immediately after the second time b is set to \(\mathrm{b/2}\) (i.e., the second time it is halved).
Solution:
- Condensed Solution Implementation:
Simulate the algorithm step-by-step, specifically counting how many times b is halved and noting the resulting value after the second such event. - Necessary Data points:
Start: \(\mathrm{a = 35, b = 27, T = 0}\). Track each step until b is halved twice. - Calculations Estimations:
Step 1: \(\mathrm{b=27}\) is odd → \(\mathrm{T = T + a = 35, b = 26, a = 70}\). Step 2: \(\mathrm{b=26}\) is even → First halving: \(\mathrm{b = 13, a = 140}\). Step 3: \(\mathrm{b=13}\) is odd → \(\mathrm{T = T + a = 35+140=175, b = 12, a = 280}\). Step 4: \(\mathrm{b=12}\) is even → Second halving: \(\mathrm{b = 6, a = 560}\). The value of b after the second halving is 6. - Comparison to Answer Choices:
Answer choices likely include 6, matching our calculation.
FINAL ANSWER Blank 2: 6
Summary
For both blanks, careful tracking of the algorithm's steps yields the answers without running the entire process. For blank 1, the first update to T happens immediately, and for blank 2, counting the two occurrences of '\(\mathrm{b = b/2}\)' gives the answer. Understanding when conditions are met—like b being odd or even—helps solve these quickly.
Question Independence Analysis
The two blanks are independent: each uses different input values and asks about different variables or steps in the algorithm. Solving one does not inform or affect solving the other.