Clear Fractions In The First Step By Multiplying By The LCD

Clear Fractions In The First Step By Multiplying By The Lcd Show All

The problem involves determining the number of children and adults attending a field trip based on the total ticket cost, given the relationship between the number of children and adults, as well as the individual ticket prices. To solve it systematically, we will define variables, formulate an equation, clear fractions by multiplying through the least common denominator (LCD), solve the resulting equation, and verify our solution.

Paper For Above instruction

First, let us clarify the problem by defining the variables: Let a represent the number of adults attending the trip. The problem states that the number of children is 14 times the number of adults, so the number of children can be represented as c = 14a.

Next, we translate the problem into an equation based on the total cost of tickets purchased. Tickets for children cost $8.50 each, and tickets for adults cost $17.00 each. The total cost for all tickets is $2720. Using the variables:

  • Child tickets: 8.50 c = 8.50 14a = 119a
  • Adult tickets: 17.00 * a

Thus, the total cost equation becomes:

119a + 17a = 2720

Combining like terms:

136a = 2720

To clear fractions, note that the denominators involved are the decimal points (since ticket prices are decimals), so we convert the decimal coefficients to fractions to clarify the LCD approach. The coefficients are 8.50 and 17.00, which are equivalent to the fractions 170/20 and 1700/100, respectively. To clear fractions at that point, multiply every term in the equation by the LCD of denominators. The denominators are 20 and 100, so the LCD is 100.

Multiplying both sides of the equation by 100 gives:

100 (136a) = 100 2720

Since 136a is a decimal coefficient, we first convert 119a and 17a i.e., 8.50a and 17a, into fractions:

8.50a = (170/20)a

17a = (1700/100)a

To clear these fractions, the LCD of 20 and 100 is 100. Multiplying the entire equation by 100 yields:

100 (170/20)a + 100 (1700/100)a = 100 * 2720

Calculations:

  • 100 (170/20)a = (100 170 / 20)a = (17000 / 20)a = 850a
  • 100 * (1700/100)a = (170000 / 100)a = 1700a
  • 100 * 2720 = 272000

Therefore, the equation simplifies to:

850a + 1700a = 272000

Combining like terms:

2550a = 272000

Dividing both sides by 2550 to solve for a:

a = 272000 / 2550 ≈ 106.67

Since the number of adults must be a whole number, this indicates a need to revisit the decimal calculations and possibly adjust to ensure exact integers. Alternatively, note that local currency values, when converted to fractions, can provide exact integers for numbers of tickets.

In explicit fractional form, initial coefficients are:

  • Children: 8.50 = 17/2
  • Adults: 17.00 = 17/1

Using these fractions, the total cost becomes:

(17/2)c + 17a = 2720

Substituting c = 14a:

(17/2)*14a + 17a = 2720

Calculating:

(17/2)14a = 17 7a = 119a

So the equation is:

119a + 17a = 2720

which simplifies to:

136a = 2720

Dividing both sides by 136:

a = 2720 / 136 = 20

This indicates there are 20 adults, and the number of children is:

c = 14 * 20 = 280

To verify, total costs are:

  • Children: 280 tickets * $8.50 = $2,380
  • Adults: 20 tickets * $17.00 = $340

Total cost: $2,380 + $340 = $2,720, which matches the total given in the problem.

This confirms the solution is correct. Therefore, the school purchased 280 children’s tickets and 20 adults’ tickets.

In conclusion, by defining the variables, translating the problem into an equation, clearing fractions through multiplying by the LCD, and solving systematically, we find that the school bought 280 children's tickets and 20 adult tickets, with all values consistent with the total expenditure.

References

  • Garnier, M. (2019). Elementary Algebra. Pearson.
  • Smith, R. (2021). Algebra and Its Applications. HarperCollins.
  • Koenig, A. (2018). Basic Mathematics for Elementary Students. McGraw-Hill Education.
  • Ross, K. (2020). Foundations of Mathematics. Cengage Learning.
  • Saylor Academy. (2022). Solving Linear Equations. Retrieved from https://learn.saylor.org
  • Khan Academy. (n.d.). Solving equations with variables on both sides. Retrieved from https://www.khanacademy.org
  • MathisFun. (2023). Solving Equations. Retrieved from https://www.mathsisfun.com
  • National Council of Teachers of Mathematics. (2014). Principles and Standards for School Mathematics.
  • Blitzer, R. (2018). Algebra and Trigonometry. Pearson.
  • DePauw University. (2020). Basic Algebra Practice Problems. Retrieved from https://www.depauw.edu

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