Example Question - fractions
Here are examples of questions we've helped users solve.
Simplifying an Algebraic Expression
<p> Primero, simplificamos la expresión dentro de los corchetes: </p> <p> \[ \left( \frac{2}{7} \right)^2 \cdot \left( -\frac{1}{6} \right)^2 = \frac{4}{49} \cdot \frac{1}{36} \] </p> <p> Multiplicamos las fracciones: </p> <p> \[ \frac{4 \cdot 1}{49 \cdot 36} = \frac{4}{1764} \] </p> <p> Finalmente, simplificamos \(\frac{4}{1764}\): </p> <p> \[ \frac{4 \div 4}{1764 \div 4} = \frac{1}{441} \] </p> <p> Por lo tanto, el resultado final es: </p> <p> \[ \left( \left( \frac{2}{7} \right)^2 \cdot \left( -\frac{1}{6} \right)^2 \right)^2 = \left( \frac{1}{441} \right)^2 = \frac{1}{194481} \] </p>
Multiplication of Fractions and Weight
<p>First, express the mixed number as an improper fraction:</p> <p>\(9 \frac{7}{6} = \frac{9 \times 6 + 7}{6} = \frac{54 + 7}{6} = \frac{61}{6}\)</p> <p>Now multiply by \(32 \, \text{kg}\):</p> <p>\(\frac{61}{6} \times 32 = \frac{61 \times 32}{6} = \frac{1952}{6} = 325.33 \, \text{kg}\)</p> <p>The final result is approximately \(325.33 \, \text{kg}\).</p>
Multiplication of Mixed Numbers and Improper Fractions
<p>Convert the mixed number to an improper fraction:</p> <p>2 \frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{7}{3}</p> <p>Now, multiply:</p> <p>\frac{7}{3} \times \frac{4 \times 3 + 3}{7} = \frac{7}{3} \times \frac{15}{7}</p> <p>Cancel the 7s:</p> <p> \frac{15}{3} = 5</p> <p>Thus, the solution is 5.</p>
Calculate the Sums
<p> Primero, encontramos un denominador común para las fracciones </p> <p> El denominador común de 4, 2 y 3 es 12. </p> <p> Convertimos las fracciones: </p> <p> \(\frac{3}{4} = \frac{9}{12}, \quad \frac{7}{2} = \frac{42}{12}, \quad \frac{7}{3} = \frac{28}{12}\) </p> <p> Luego, sumamos las fracciones: </p> <p> \(\frac{9}{12} + \frac{42}{12} + \frac{28}{12} = \frac{79}{12}\) </p> <p> La respuesta es \(\frac{79}{12}\). </p>
Calculate the operations
<p>Primero, encontramos un denominador común para las fracciones. El común denominador de \(3\) y \(4\) es \(12\).</p> <p>Convertimos las fracciones: \(\frac{2}{3} = \frac{8}{12}\), \(\frac{7}{3} = \frac{28}{12}\), y \(\frac{1}{4} = \frac{3}{12}\).</p> <p>Ahora, realizamos la operación: \(\frac{8}{12} + \frac{28}{12} - \frac{3}{12} = \frac{8 + 28 - 3}{12} = \frac{33}{12}\).</p> <p>Finalmente, simplificamos: \(\frac{33}{12} = \frac{11}{4}\).</p>
Calculating Fractions
<p>a) \frac{2}{3} + \frac{7}{3} - \frac{1}{4} = \frac{2 + 7 \cdot 1 - \frac{3}{4}}{3} = \frac{9 - 0.75}{3} = \frac{8.25}{3} = \frac{33}{12} = \frac{11}{4}</p> <p>b) 2 \frac{1}{3} + \frac{1}{2} - \frac{1}{5} - \frac{1}{15} = \frac{7}{3} + \frac{1}{2} - \frac{1}{5} - \frac{1}{15} = \frac{70 + 15 - 6 - 2}{30} = \frac{77}{30}</p> <p>c) \frac{13}{36} - \frac{5}{4} - \frac{2}{9} = \frac{13}{36} - \frac{45}{36} - \frac{8}{36} = \frac{13 - 45 - 8}{36} = \frac{-40}{36} = -\frac{10}{9}</p> <p>d) 3 + \left( -\frac{5}{4} \right) + 2 \frac{3}{4} = 3 - \frac{5}{4} + \frac{11}{4} = 3 + \frac{6}{4} = 3 + \frac{3}{2} = \frac{6 + 3}{2} = \frac{9}{2}</p> <p>e) \frac{5}{6} - \frac{2}{9} = \frac{15}{18} - \frac{4}{18} = \frac{11}{18}</p> <p>f) 7 \frac{1}{4} - \left( 4 - \frac{1}{2} \right) = \frac{29}{4} - \left( \frac{8}{2} - \frac{1}{2} \right) = \frac{29}{4} - \frac{7}{2} = \frac{29 - 14}{4} = \frac{15}{4}</p>
Simplifying a Mathematical Expression
<p>First, express each term correctly:</p> <p>1. Convert mixed number \(1 \frac{296}{36}\) to improper fraction: \( \frac{296 + 36}{36} = \frac{332}{36} \)</p> <p>2. Perform the operation \( \frac{332}{36} \div 36 \): this becomes \( \frac{332}{36 \times 36} = \frac{332}{1296} \)</p> <p>3. Multiply by \( \frac{1}{216} \): \( \frac{332}{1296} \times \frac{1}{216} = \frac{332}{1296 \times 216} \)</p> <p>4. Finally, simplify \( \frac{332}{1296 \times 216} \) to lowest terms if possible.</p>
Simplifying a Fractional Expression
<p>First, simplify the denominator: </p> <p>(8x²y³)³ = 8³(x²)³(y³)³ = 512x^{6}y^{9}</p> <p>Now, the expression becomes:</p> <p>\(\frac{50x^{-3}y^{14}}{512x^{6}y^{9}}\)</p> <p>Next, simplify by dividing the coefficients and subtracting the exponents:</p> <p>\(=\frac{50}{512}x^{-3-6}y^{14-9}\)</p> <p>Which simplifies to:</p> <p>\(\frac{25}{256}x^{-9}y^{5}\)</p> <p>Finally, rewrite with positive exponents:</p> <p>\(\frac{25y^{5}}{256x^{9}}\)</p>
Simplifying Mathematical Expressions
<p>First, convert mixed numbers to improper fractions:</p> <p>For \(1 \frac{296}{3}\), it becomes \(\frac{3 \times 1 + 296}{3} = \frac{299}{3}\).</p> <p>For \(36 \frac{5}{2}\), it becomes \(\frac{2 \times 36 + 5}{2} = \frac{77}{2}\).</p> <p>Now, substitute into the expression:</p> <p>\(\frac{299}{3} \div \frac{77}{2} \times \frac{1}{216}\).</p> <p>Dividing by a fraction is equivalent to multiplying by its reciprocal:</p> <p>\(\frac{299}{3} \times \frac{2}{77} \times \frac{1}{216} = \frac{299 \times 2}{3 \times 77 \times 216}\).</p> <p>Now, calculate the numerator and the denominator:</p> <p>Numerator: \(299 \times 2 = 598\).</p> <p>Denominator: \(3 \times 77 \times 216 = 5004\).</p> <p>So, the expression simplifies to:</p> <p>\(\frac{598}{5004}\).</p> <p>This can be simplified further by finding the GCD of 598 and 5004, which is 2:</p> <p>Thus, \(\frac{598 \div 2}{5004 \div 2} = \frac{299}{2502}\).</p> <p>The final simplified result is \(\frac{299}{2502}\).</p>
Comparing Rational Numbers
<p>Pour comparer les nombres rationnels, nous allons évaluer chaque paire en les ramenant à un même dénominateur ou en calculant leur valeur décimale.</p> <p>Cas 1: Comparer \( \frac{43}{34} \) et \( \frac{27}{34} \). Comme ils ont le même dénominateur, \( 43 > 27 \), donc \( \frac{43}{34} > \frac{27}{34} \).</p> <p>Cas 2: Comparer \( \frac{43}{47} \) et \( \frac{2}{35} \). Calculons les valeurs décimales: \( \frac{43}{47} \approx 0.915 \) et \( \frac{2}{35} \approx 0.057 \), donc \( \frac{43}{47} > \frac{2}{35} \).</p> <p>Cas 3: Comparer \( \frac{23}{57} \) et \( \frac{17}{38} \). Calculons les valeurs décimales: \( \frac{23}{57} \approx 0.404 \) et \( \frac{17}{38} \approx 0.447 \), donc \( \frac{23}{57} < \frac{17}{38} \).</p> <p>Cas 4: Comparer \( \frac{34}{45} \) et \( \frac{41}{60} \). Calculons les valeurs décimales: \( \frac{34}{45} \approx 0.756 \) et \( \frac{41}{60} \approx 0.683 \), donc \( \frac{34}{45} > \frac{41}{60} \).</p>
Division and Multiplication of Fractions
<p>To solve the expression \( \frac{2}{4} \div \frac{2}{16} \), we first rewrite the division as multiplication by the reciprocal:</p> <p>\( \frac{2}{4} \times \frac{16}{2} \)</p> <p>Now, we can simplify:</p> <p>\( \frac{2 \times 16}{4 \times 2} = \frac{32}{8} \)</p> <p>Next, simplify \( \frac{32}{8} \):</p> <p>Thus, the final answer is \( 4 \).</p>
Arithmetic Division of Fractions
<p>لحل القسمة بين الكسرين, نقلب الكسر الثاني ونضربه بالكسر الأول.</p> <p>\[ \frac{35}{33} \div \frac{9}{22} = \frac{35}{33} \times \frac{22}{9} \]</p> <p>نضرب البسط في البسط والمقام في المقام.</p> <p>\[ = \frac{35 \times 22}{33 \times 9} \]</p> <p>الآن نبسط الكسور قبل الضرب إذا أمكن ذلك.</p> <p>\[ = \frac{35}{3} \times \frac{2}{9} \]</p> <p>نضرب البسط في البسط والمقام في المقام.</p> <p>\[ = \frac{35 \times 2}{3 \times 9} \]</p> <p>\[ = \frac{70}{27} \]</p> <p>نحصل على النتيجة وهي \(\frac{70}{27}\).</p>
Understanding a Series of Fractions
<p>Para resolver la serie de fracciones, se suma cada término dado:</p> <p>\[ \frac{1}{2} + \frac{1}{3} + \frac{1}{6} \]</p> <p>Para sumar fracciones, se necesitan denominadores comunes. El mínimo común denominador (MCD) de 2, 3 y 6 es 6.</p> <p>\[ \frac{3}{6} + \frac{2}{6} + \frac{1}{6} \]</p> <p>Ahora que todas las fracciones tienen el mismo denominador, se pueden sumar los numeradores:</p> <p>\[ \frac{3 + 2 + 1}{6} \]</p> <p>\[ \frac{6}{6} \]</p> <p>Finalmente, se simplifica la fracción:</p> <p>\[ \frac{6}{6} = 1 \]</p>
Simplifying Rational Expressions
<p>Для решения данной задачи необходимо упростить рациональное выражение, содержащееся в восьмом номере:</p> <p>\[ \frac{a-1}{2} + \frac{3a-1}{4} - \frac{5a-1}{6} \]</p> <p>Приведём дроби к общему знаменателю, который будет равен 12:</p> <p>\[ \frac{6(a-1)}{12} + \frac{3(3a-1)}{12} - \frac{2(5a-1)}{12} \]</p> <p>Раскроем скобки в числителях:</p> <p>\[ \frac{6a - 6}{12} + \frac{9a - 3}{12} - \frac{10a - 2}{12} \]</p> <p>Теперь сложим дроби, объединив числители:</p> <p>\[ \frac{6a - 6 + 9a - 3 - 10a + 2}{12} \]</p> <p>Произведём сложение и вычитание чисел в числителе:</p> <p>\[ \frac{6a + 9a - 10a - 6 - 3 + 2}{12} \]</p> <p>\[ \frac{5a - 7}{12} \]</p> <p>Итак, упрощённое рациональное выражение:</p> <p>\[ \frac{5a - 7}{12} \]</p>
Converting Fractions to Quotients and Unit Conversions in Mathematics
Für die erste Frage (2): <p>a) $\frac{24}{52} = \frac{6}{13}$, nach der Kürzung durch 4</p> <p>b) $\frac{24}{16} = \frac{3}{2}$, nach der Kürzung durch 8</p> <p>c) $\frac{24}{18} = \frac{4}{3}$, nach der Kürzung durch 6</p> <p>d) $\frac{24}{36} = \frac{2}{3}$, nach der Kürzung durch 12</p> <p>e) $\frac{24}{84} = \frac{2}{7}$, nach der Kürzung durch 12</p> Für die zweite Frage (3): <p>a) $\frac{3}{4}h$ (in min) = $\frac{3}{4} \cdot 60\,min = 45\,min$</p> <p>b) $\frac{3}{5}m$ (in mm) = $\frac{3}{5} \cdot 1000\,mm = 600\,mm$</p> <p>c) $\frac{14}{20}ha$ (in a) = $\frac{14}{20} \cdot 100\,a = 70\,a$</p> <p>d) $\frac{6}{125}kg$ (in g) = $\frac{6}{125} \cdot 1000\,g = 48\,g$</p> <p>e) $\frac{3}{4}km$ (in m) = $\frac{3}{4} \cdot 1000\,m = 750\,m$</p>
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com