this series of problems will build your fluency with considering overlapping elements when analyzing part…

this series of problems will build your fluency with considering overlapping elements when analyzing part - whole relationships.\n- use the answer key to determine that you accurately solved the problem.\n- if necessary, revise your work.\n1. a group of 46 people were surveyed about their participation in 3 hobbies: reading, hiking, and cooking. everyone participates in at least 1 hobby. 21 people enjoy reading, 23 people enjoy hiking, and 24 people enjoy cooking. 5 people enjoy reading and hiking but not cooking. 3 people enjoy hiking and cooking but not reading. there are 3 people who enjoy all 3 hobbies. how many people enjoy reading and cooking but not hiking?

Answer

Explanation:

Step1: Define variables for each region

Let ( R ) be reading, ( H ) be hiking, ( C ) be cooking. Let ( x ) be the number of people who enjoy reading and cooking but not hiking. The number of people who enjoy only reading is ( 21 - 5 - 3 - x ), only hiking is ( 23 - 5 - 3 - 3 ), only cooking is ( 24 - x - 3 - 3 ), reading and hiking only is 5, hiking and cooking only is 3, all three is 3, and reading and cooking only is ( x ).

Step2: Sum all regions to total

The total number of people is 46. So: [ (21 - 5 - 3 - x) + (23 - 5 - 3 - 3) + (24 - x - 3 - 3) + 5 + 3 + x + 3 = 46 ] Simplify each term:

Only reading: ( 13 - x )
Only hiking: ( 9 )
Only cooking: ( 15 - x ) Now sum all terms: [ (13 - x) + 9 + (15 - x) + 5 + 3 + x + 3 = 46 ] Combine like terms: [ 13 - x + 9 + 15 - x + 5 + 3 + x + 3 = 46 \ (13 + 9 + 15 + 5 + 3 + 3) + (-x - x + x) = 46 \ 48 - x = 46 ]

Step3: Solve for ( x )

Subtract 48 from both sides: [ -x = 46 - 48 \ -x = -2 \ x = 2 ]