Найти значение переменной x, являющееся решением уравнения: 3/7 : 1 1/14 = 0,4 : (4-x)

Привет! Давай разберемся с этим уравнением вместе.

Дано:

• \[ \frac{3}{7} : 1\frac{1}{14} = 0,4 : (4-x) \]

Найти:

• \[ x \]

Решение:

1. Шаг 1: Преобразуем смешанное число и десятичную дробь в обыкновенные.
   • Смешанное число
     The problem asks to find the value of the variable 'x' which is the solution to the given equation. The equation is: `3/7 : 1 1/14 = 0.4 : (4-x)`
   • Step 1: Convert the mixed number and the decimal to fractions.
     • Mixed number
       The mixed number is $$1\frac{1}{14}$$. To convert it to an improper fraction, we multiply the whole number part (1) by the denominator (14) and add the numerator (1). So, $$1 \times 14 + 1 = 15$$. The denominator remains 14. Thus, $$1\frac{1}{14} = \frac{15}{14}$$.
     • Decimal number: $$0.4$$. To convert it to a fraction, we write it as 4/10, which can be simplified to 2/5.
   • Step 2: Rewrite the equation with the converted fractions.
     • The equation now becomes:
       The equation is $$\frac{3}{7} : \frac{15}{14} = \frac{2}{5} : (4-x)$$.
   • Step 3: Perform the division on the left side of the equation.
     • Division by a fraction is the same as multiplying by its reciprocal. So, $$\frac{3}{7} : \frac{15}{14} = \frac{3}{7} \times \frac{14}{15}$$.
     • We can simplify this multiplication:
       $$\frac{3}{7} \times \frac{14}{15} = \frac{3 \times 14}{7 \times 15}$$.
     • We can cancel out common factors. 7 goes into 14 twice, and 3 goes into 15 five times.
     • So, $$\frac{3 \times 14}{7 \times 15} = \frac{1 \times 2}{1 \times 5} = \frac{2}{5}$$.
   • Step 4: Rewrite the equation with the simplified left side.
     • The equation is now:
       $$\frac{2}{5} = \frac{2}{5} : (4-x)$$.
   • Step 5: Solve for (4-x).
     • From the equation $$\frac{2}{5} = \frac{2}{5} : (4-x)$$, we can see that if we divide $$\frac{2}{5}$$ by some number and get $$\frac{2}{5}$$, that number must be 1.
     • So, $$(4-x)$$ must be equal to 1.
     • \[ 4-x = 1 \]
   • Step 6: Solve for x.
     • To find x, subtract 1 from 4:
     • \[ x = 4 - 1 \]
     • \[ x = 3 \]
   • Step 7: Verify the solution.
     • Substitute x = 3 back into the original equation:
     • Left side: $$\frac{3}{7} : 1\frac{1}{14} = \frac{3}{7} : \frac{15}{14} = \frac{3}{7} \times \frac{14}{15} = \frac{2}{5}$$.
     • Right side: $$0,4 : (4-3) = 0,4 : 1 = 0,4 = \frac{2}{5}$$.
     • Since both sides are equal, our solution is correct.

Ответ: x = 3