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Difficulty: MediumRatios, Rates, and Proportions

A youth athletic league has 140 total players registered for either soccer or basketball. Initially, the ratio of soccer players to basketball players is 4:34 : 3. If 15 additional basketball players join the league and no soccer players leave, what is the new ratio of soccer players to basketball players in simplest form?

  1. A
    8:158 : 15
  2. 16:1516 : 15Answer
  3. C
    4:54 : 5
  4. D
    16:1916 : 19
  5. E
    3:53 : 5

Answer

The new ratio of soccer players to basketball players in simplest form is 16:1516 : 15.
The initial ratio of 4:34 : 3 breaks 140 players into 7 equal parts of 20 players each, yielding 80 soccer players and 60 basketball players. Adding 15 basketball players gives 75 basketball players. The ratio of 80 soccer players to 75 basketball players simplifies to 16:1516 : 15.

Step-by-Step Solution

1
Determine the initial number of soccer and basketball players using the given ratio and total.
The total ratio parts equal 4+3=74 + 3 = 7. Value per part =140/7=20= 140 / 7 = 20. Soccer players =4×20=80= 4 \times 20 = 80, Basketball players =3×20=60= 3 \times 20 = 60.
Converting ratio parts into actual counts is required before adjusting player totals.
2
Add 15 players to the basketball total.
New basketball player count =60+15=75= 60 + 15 = 75. Soccer player count remains 8080.
The problem states 15 new basketball players joined while soccer player count remained constant.
3
Form the new ratio of soccer players to basketball players and simplify.
80:75=805:755=16:1580 : 75 = \frac{80}{5} : \frac{75}{5} = 16 : 15.
Dividing both quantities by their greatest common factor, 5, expresses the ratio in simplest terms.

Key Concept

Solving multi-step ratio problems by determining actual quantities from a given total and updating quantities following a change.
Estimated Time:1m 30s
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Topics

Ratios, Rates, and Proportions

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