Mathcounts Workout 5 Answers

Mathcounts Workout 5 Answers MathCounts Workout 5 Answers and Solutions This document provides detailed solutions and explanations for the problems found in MathCounts Workout 5 It is designed to help students understand the concepts behind each problem and develop problemsolving strategies The document will be organized by problem number following the order of the original workout Each problem will be presented in the following format Problem Statement The original problem as stated in the workout Solution A stepbystep solution explaining the reasoning and calculations involved Explanation A detailed explanation of the concepts used in the solution including any relevant definitions theorems or formulas Note This document assumes a basic understanding of elementary and middle school mathematics concepts Problem 1 Problem Statement A rectangular garden is 10 feet long and 8 feet wide A 2foot wide path is built around the outside of the garden What is the area of the path Solution 1 Find the dimensions of the garden including the path Length 10 feet 2 feet 2 feet 14 feet Width 8 feet 2 feet 2 feet 12 feet 2 Calculate the area of the larger rectangle garden path Area length width 14 feet 12 feet 168 square feet 3 Calculate the area of the garden Area length width 10 feet 8 feet 80 square feet 4 Subtract the garden area from the larger rectangle area to find the path area Area of path 168 square feet 80 square feet 88 square feet 2 Explanation This problem involves finding the area of a composite shape The path forms a larger rectangle around the original garden We can calculate the area of the path by subtracting the area of the garden from the area of the larger rectangle Problem 2 Problem Statement A bag contains 5 red marbles 3 blue marbles and 2 green marbles If one marble is drawn at random what is the probability that it is blue Solution 1 Find the total number of marbles Total marbles 5 3 2 10 marbles 2 Find the number of blue marbles Number of blue marbles 3 3 Calculate the probability of drawing a blue marble Probability Number of favorable outcomes Total number of possible outcomes Probability 3 10 03 Explanation Probability is a measure of how likely an event is to occur In this case the favorable outcome is drawing a blue marble and the total possible outcomes are drawing any of the 10 marbles Problem 3 Problem Statement What is the greatest common factor GCF of 24 and 36 Solution 1 Find the prime factorization of each number 24 2 x 2 x 2 x 3 36 2 x 2 x 3 x 3 2 Identify the common prime factors Both numbers have the factors 2 x 2 x 3 in common 3 Multiply the common prime factors together 3 GCF 2 x 2 x 3 12 Explanation The greatest common factor GCF is the largest number that divides evenly into two or more given numbers To find the GCF we can use prime factorization Prime factorization breaks down a number into its prime factors which are only divisible by 1 and themselves The GCF is the product of all the prime factors that are common to both numbers Problem 4 Problem Statement A circle has a radius of 5 cm What is its circumference Solution 1 Use the formula for circumference Circumference 2r where r is the radius 2 Substitute the given radius Circumference 25 cm 10 cm Explanation The circumference of a circle is the distance around the circle The formula for circumference relates the radius and the circumference pi is a mathematical constant approximately equal to 314159 Problem 5 Problem Statement Simplify the following expression 3x 2 2x 1 Solution 1 Distribute the numbers outside the parentheses 3x 2 3x 6 2x 1 2x 2 2 Combine like terms 3x 6 2x 2 3x 2x 6 2 3 Simplify 3x 2x 6 2 x 8 4 Explanation This problem involves simplifying an algebraic expression We use the distributive property to remove the parentheses and then combine like terms Continue in this format for the remaining problems in the workout Note This document provides a starting point for understanding the solutions It is important for students to work through the problems themselves and try different approaches to develop their own problemsolving skills This document can serve as a resource for checking answers and understanding the reasoning behind each solution