Question
\( \begin{array}{lll}\text { a. } \frac{5}{6}-\frac{3}{4}: \frac{2}{5}+\frac{3}{2} & \text { c. } \frac{5}{6}-\frac{3}{4} \cdot \frac{2}{5}+\frac{3}{2} & \text { e. } \frac{1}{2} \cdot \frac{3}{4}: \frac{3}{5}+\frac{3}{2} \\ \text { b. } 6-\frac{3}{8}: \frac{5}{4}-\frac{1}{20} & \text { d. } \frac{1}{2}: \frac{3}{4} \cdot \frac{3}{5}+\frac{3}{2} & \text { f. } \frac{3}{4}: \frac{2}{5}-\frac{1}{3} \cdot \frac{4}{9}\end{array} \)
Ask by Stewart Peters. in Spain
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
a. \( \frac{11}{24} \)
b. \( \frac{113}{20} \) o \( 5 \frac{13}{20} \)
c. \( \frac{61}{30} \) o \( 2 \frac{1}{30} \)
d. \( \frac{47}{18} \) o \( 2 \frac{11}{18} \)
e. \( \frac{17}{8} \) o \( 2 \frac{1}{8} \)
f. \( \frac{405}{136} \) o aproximadamente \( 2.976 \)
Solution
Claro, resolveremos cada una de las expresiones proporcionadas paso a paso en español.
### a. \( \frac{5}{6} - \frac{3}{4} : \frac{2}{5} + \frac{3}{2} \)
**Paso 1:** Interpretar el símbolo ":" como división.
\[
\frac{5}{6} - \left( \frac{3}{4} \div \frac{2}{5} \right) + \frac{3}{2}
\]
**Paso 2:** Realizar la división.
\[
\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8}
\]
**Paso 3:** Sustituir y simplificar.
\[
\frac{5}{6} - \frac{15}{8} + \frac{3}{2}
\]
**Paso 4:** Encontrar el común denominador (24).
\[
\frac{5}{6} = \frac{20}{24}, \quad \frac{15}{8} = \frac{45}{24}, \quad \frac{3}{2} = \frac{36}{24}
\]
**Paso 5:** Realizar las operaciones.
\[
\frac{20}{24} - \frac{45}{24} + \frac{36}{24} = \frac{11}{24}
\]
**Respuesta:** \( \frac{11}{24} \)
---
### b. \( 6 - \frac{3}{8} : \frac{5}{4} - \frac{1}{20} \)
**Paso 1:** Realizar la división.
\[
\frac{3}{8} \div \frac{5}{4} = \frac{3}{8} \times \frac{4}{5} = \frac{12}{40} = \frac{3}{10}
\]
**Paso 2:** Simplificar la expresión.
\[
6 - \frac{3}{10} - \frac{1}{20}
\]
**Paso 3:** Encontrar el común denominador (20).
\[
6 = \frac{120}{20}, \quad \frac{3}{10} = \frac{6}{20}, \quad \frac{1}{20} = \frac{1}{20}
\]
**Paso 4:** Realizar las operaciones.
\[
\frac{120}{20} - \frac{6}{20} - \frac{1}{20} = \frac{113}{20} = 5 \frac{13}{20}
\]
**Respuesta:** \( \frac{113}{20} \) o \( 5 \frac{13}{20} \)
---
### c. \( \frac{5}{6} - \frac{3}{4} \cdot \frac{2}{5} + \frac{3}{2} \)
**Paso 1:** Realizar la multiplicación.
\[
\frac{3}{4} \times \frac{2}{5} = \frac{6}{20} = \frac{3}{10}
\]
**Paso 2:** Sustituir y simplificar.
\[
\frac{5}{6} - \frac{3}{10} + \frac{3}{2}
\]
**Paso 3:** Encontrar el común denominador (30).
\[
\frac{5}{6} = \frac{25}{30}, \quad \frac{3}{10} = \frac{9}{30}, \quad \frac{3}{2} = \frac{45}{30}
\]
**Paso 4:** Realizar las operaciones.
\[
\frac{25}{30} - \frac{9}{30} + \frac{45}{30} = \frac{61}{30} = 2 \frac{1}{30}
\]
**Respuesta:** \( \frac{61}{30} \) o \( 2 \frac{1}{30} \)
---
### d. \( \frac{1}{2} : \left( \frac{3}{4} \cdot \frac{3}{5} \right) + \frac{3}{2} \)
**Paso 1:** Realizar la multiplicación dentro del paréntesis.
\[
\frac{3}{4} \times \frac{3}{5} = \frac{9}{20}
\]
**Paso 2:** Realizar la división.
\[
\frac{1}{2} \div \frac{9}{20} = \frac{1}{2} \times \frac{20}{9} = \frac{20}{18} = \frac{10}{9}
\]
**Paso 3:** Sustituir y simplificar.
\[
\frac{10}{9} + \frac{3}{2}
\]
**Paso 4:** Encontrar el común denominador (18).
\[
\frac{10}{9} = \frac{20}{18}, \quad \frac{3}{2} = \frac{27}{18}
\]
**Paso 5:** Realizar las operaciones.
\[
\frac{20}{18} + \frac{27}{18} = \frac{47}{18} = 2 \frac{11}{18}
\]
**Respuesta:** \( \frac{47}{18} \) o \( 2 \frac{11}{18} \)
---
### e. \( \frac{1}{2} \cdot \frac{3}{4} : \frac{3}{5} + \frac{3}{2} \)
**Paso 1:** Realizar la multiplicación.
\[
\frac{1}{2} \times \frac{3}{4} = \frac{3}{8}
\]
**Paso 2:** Realizar la división.
\[
\frac{3}{8} \div \frac{3}{5} = \frac{3}{8} \times \frac{5}{3} = \frac{15}{24} = \frac{5}{8}
\]
**Paso 3:** Sustituir y simplificar.
\[
\frac{5}{8} + \frac{3}{2}
\]
**Paso 4:** Encontrar el común denominador (8).
\[
\frac{5}{8} = \frac{5}{8}, \quad \frac{3}{2} = \frac{12}{8}
\]
**Paso 5:** Realizar las operaciones.
\[
\frac{5}{8} + \frac{12}{8} = \frac{17}{8} = 2 \frac{1}{8}
\]
**Respuesta:** \( \frac{17}{8} \) o \( 2 \frac{1}{8} \)
---
### f. \( \frac{3}{4} : \left( \frac{2}{5} - \frac{1}{3} \cdot \frac{4}{9} \right) \)
**Paso 1:** Realizar la multiplicación dentro del paréntesis.
\[
\frac{1}{3} \times \frac{4}{9} = \frac{4}{27}
\]
**Paso 2:** Realizar la resta dentro del paréntesis.
\[
\frac{2}{5} - \frac{4}{27}
\]
**Paso 3:** Encontrar el común denominador (135).
\[
\frac{2}{5} = \frac{54}{135}, \quad \frac{4}{27} = \frac{20}{135}
\]
**Paso 4:** Realizar la resta.
\[
\frac{54}{135} - \frac{20}{135} = \frac{34}{135}
\]
**Paso 5:** Realizar la división.
\[
\frac{3}{4} \div \frac{34}{135} = \frac{3}{4} \times \frac{135}{34} = \frac{405}{136} = \frac{405}{136}
\]
**Respuesta:** \( \frac{405}{136} \) o aproximadamente \( 2.976 \)
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
Let's break down a couple of these expressions for better understanding! First off, when dealing with fractions, it's crucial to remember the order of operations (often remembered as PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Start by simplifying terms and ensuring common denominators where needed. For example, in expression a, tackle the subtraction first before dividing by the next fraction. Additionally, there's a handy tip for dealing with division by fractions: multiply by the reciprocal! So in expressions that require division, like \( \frac{a}{b} : \frac{c}{d} \), you can transform it to \( \frac{a}{b} \times \frac{d}{c} \). This can really save you some time and make the calculations smoother! Keep practicing, and soon working with these fractions will feel like second nature!
