I thought of a number, multiplied it by 2 1/2 , divided the result by 1 1/5 , subtracted 7/18 from it, and got 1 5/6

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I thought of a number, multiplied it by 2 1/2 , divided the result by 1 1/5 , subtracted 7/18 from it, and got 1 5/6 . What was my number?

The Correct Answer and Explanation is:

Let the number you thought of be x.

Step-by-step:

You:

  • Multiplied it by 2½ → x×52x \times \frac{5}{2}
  • Divided the result by 1⅕ → x⋅5265=x⋅52⋅56=x⋅2512\frac{x \cdot \frac{5}{2}}{\frac{6}{5}} = x \cdot \frac{5}{2} \cdot \frac{5}{6} = x \cdot \frac{25}{12}
  • Subtracted 7/18 from it: x⋅2512−718x \cdot \frac{25}{12} – \frac{7}{18}
  • The result was 1⅚, which is: 116\frac{11}{6}

Equation:

x⋅2512−718=116x \cdot \frac{25}{12} – \frac{7}{18} = \frac{11}{6}

Solve step-by-step:

  1. Add 718\frac{7}{18} to both sides: x⋅2512=116+718x \cdot \frac{25}{12} = \frac{11}{6} + \frac{7}{18}
  2. Get common denominator (LCM of 6 and 18 is 18): 3318+718=4018=209\frac{33}{18} + \frac{7}{18} = \frac{40}{18} = \frac{20}{9}
  3. So: x⋅2512=209x \cdot \frac{25}{12} = \frac{20}{9}
  4. Multiply both sides by the reciprocal of 2512\frac{25}{12}: x=209⋅1225=240225=1615x = \frac{20}{9} \cdot \frac{12}{25} = \frac{240}{225} = \frac{16}{15}

✅ Final Answer:

1615\boxed{\frac{16}{15}}

Explanation

To solve this type of algebraic problem, we begin by translating each step of the word problem into a mathematical expression. The unknown number is represented by x. You are told that the number was first multiplied by 2½, or 52\frac{5}{2}, and then the result was divided by 1⅕, which is 65\frac{6}{5}.

When dividing by a fraction, we multiply by its reciprocal. So: x×52÷65=x⋅52⋅56=x⋅2512x \times \frac{5}{2} \div \frac{6}{5} = x \cdot \frac{5}{2} \cdot \frac{5}{6} = x \cdot \frac{25}{12}

Next, you subtract 718\frac{7}{18} from this result. We are told this final expression equals 116\frac{11}{6}, which is the improper fraction form of 1⅚. This gives the equation: x⋅2512−718=116x \cdot \frac{25}{12} – \frac{7}{18} = \frac{11}{6}

Solving this equation requires first isolating the term with x by adding 718\frac{7}{18} to both sides. This gives: x⋅2512=116+718x \cdot \frac{25}{12} = \frac{11}{6} + \frac{7}{18}

With a common denominator, these become: 3318+718=4018=209\frac{33}{18} + \frac{7}{18} = \frac{40}{18} = \frac{20}{9}

Then, to isolate x, divide both sides by 2512\frac{25}{12}, or multiply by its reciprocal: x=209⋅1225=240225=1615x = \frac{20}{9} \cdot \frac{12}{25} = \frac{240}{225} = \frac{16}{15}

Thus, the number you originally thought of was 1615\boxed{\frac{16}{15}}.

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