Solving Equations Using All Methods: Worded problems

To solve this problem, we will find a common equation to account for all students:

• Activity: Playing catch, Fraction: $\frac{1}{5}x$
• Activity: Playing soccer, Fraction: 20% of $x$ which is $\frac{1}{5}x$
• Activity: Watching a movie, Number: $15$ students

The total number of students involved is $x$. Thus, the setup for the equation is:

$\frac{1}{5}x + \frac{1}{5}x + 15 = x$

Simplify and solve for $x$:

$\frac{2}{5}x + 15 = x$

Subtract $\frac{2}{5}x$ from both sides:

$15 = x - \frac{2}{5}x$

$15 = \frac{3}{5}x$

To isolate $x$, multiply both sides by $\frac{5}{3}$:

$x = 15 \times \frac{5}{3}$

$x = 25$

Therefore, the total number of students is 25 students.

To solve this problem, let's determine how many runs Roberto did on each day and then sum those amounts.

Starting with the given conditions:

• On the first day, Roberto performs $3$ runs.

• On the second day, Roberto performs $3 + 2 = 5$ runs (2 more than on the first day).

• On the third day, Roberto performs $5 + 3 = 8$ runs (3 more than on the second day).

• On the fourth day, Roberto performs $8 + 5 = 13$ runs (5 more than on the third day).

Now, let's sum up the runs from all four days:

$\text{Total runs} = \text{First day} + \text{Second day} + \text{Third day} + \text{Fourth day} \\ = 3 + 5 + 8 + 13 \\ = 29$

Therefore, Roberto did a total of 29 runs over the four days.

The correct answer is: $(3)$ 29 runs.

To solve this problem, we'll follow these steps:

• Step 1: Define variables based on the problem.
• Step 2: Set up an equation using the given total and ratio.
• Step 3: Solve the equation to find the number of girls.
• Step 4: Calculate the number of boys using the ratio.

Now, let's work through each step:
Step 1: Let $g$ be the number of girls. According to the problem, there are 3 times as many boys as girls, so the number of boys is $3g$.
Step 2: Since the total number of students is 28, we set up the equation:
$\ g + 3g = 28$
Step 3: Combine like terms to simplify the equation:
$\ 4g = 28$
To solve for $g$, divide both sides by 4:
$\ g = \frac{28}{4} = 7$
Step 4: Calculate the number of boys as $3g$:
$\ 3g = 3 \times 7 = 21$

Therefore, the solution to the problem is 21 boys in the class.

To solve this problem, follow these steps:

• Step 1: Define the total number of dolls in the toy shop as $x$.
• Step 2: Note that 30% of these dolls are standard issue, thus $0.30x$ are standard issue dolls.
• Step 3: Since 70% of the dolls are limited edition (as standard and limited edition must account for 100% of the shop's dolls), $0.70x$ would be limited edition dolls.
• Step 4: Set up the equation: $0.70x = 21$, since we know the exact count of limited edition dolls is 21.
• Step 5: Solve for $x$ by dividing both sides of the equation by 0.70:

Therefore, the total number of dolls in the shop is $30$.

To solve this problem, follow these steps:

• Define the variables: Let $x$ be the weight of oranges in the blue box.

• Set up the equation: Since the red box has 5 kg more than the blue box, its weight is $x + 5$. The total weight of the two boxes is given as $61 \frac{1}{2}$ kg. Thus, the equation is:

$x + (x + 5) = 61 \frac{1}{2}$

Now, simplify and solve the equation step by step:

• Combine like terms: $2x + 5 = 61 \frac{1}{2}$

• Convert the mixed number to an improper fraction for easier calculations: $61 \frac{1}{2} = \frac{123}{2}$

• Write the equation with the fraction: $2x + 5 = \frac{123}{2}$

• Subtract 5 from both sides: $2x = \frac{123}{2} - 5$

• Convert 5 to a fraction with the same denominator: $5 = \frac{10}{2}$

• Subtract the fractions: $2x = \frac{123}{2} - \frac{10}{2} = \frac{113}{2}$

• Divide both sides by 2 to solve for $x$: $x = \frac{113}{2} \div 2 = \frac{113}{4}$

Thus, the weight of oranges in the blue box is $x = \frac{113}{4} = 28 \frac{1}{4}$ kg.

The red box's oranges weigh $x + 5 = \frac{113}{4} + \frac{20}{4} = \frac{133}{4} = 33 \frac{1}{4}$ kg.

Therefore, the solution is:

^{blue box $28\frac{1}{4}$ red box $33\frac{1}{4}$}