The ratio of the number of children to the number of adults at an exhibition is $7: 2.26$ children leave the exhibition. $20$ more adults arrive at the exhibition. The ratio of the number of children to the number of adults then becomes $1: 2$ . How many people are there at the exhibition at first?

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Solve a system of equations using substitution: word problems

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Q. The ratio of the number of children to the number of adults at an exhibition is

7: 2.26

children leave the exhibition.

20

more adults arrive at the exhibition. The ratio of the number of children to the number of adults then becomes

1: 2

. How many people are there at the exhibition at first?

Denote Children and Adults: Let's denote the number of children as $C$ and the number of adults as $A$ . According to the problem, the initial ratio of children to adults is $7:2$ . Therefore, we can write the relationship as $\frac{C}{A} = \frac{7}{2}$ .
Express Numbers with Ratio: From the ratio, we can express the number of children as $C = 7k$ and the number of adults as $A = 2k$ , where $k$ is a positive integer that will scale the ratio to the actual numbers of children and adults.
Changes in Numbers: According to the problem, $26$ children leave the exhibition, and $20$ more adults arrive. This changes the number of children to $C - 26$ and the number of adults to $A + 20$ .
New Ratio Equation: After these changes, the new ratio of the number of children to the number of adults becomes $1:2$ . This gives us a new equation: $\frac{C - 26}{A + 20} = \frac{1}{2}$ .
Substitute and Solve: Now we have two equations and two unknowns. The first equation is $C = 7k$ and $A = 2k$ , and the second equation is $\frac{C - 26}{A + 20} = \frac{1}{2}$ . We can substitute $C$ and $A$ from the first equation into the second equation to find the value of $k$ .
Cross-Multiply to Eliminate Fraction: Substituting $C$ and $A$ into the second equation, we get $\frac{7k - 26}{2k + 20} = \frac{1}{2}$ . To solve for $k$ , we cross-multiply to get rid of the fraction: $2(7k - 26) = 1(2k + 20)$ .
Simplify and Solve for k: Simplifying the equation, we get $14k - 52 = 2k + 20$ . Now, we will solve for $k$ by first subtracting $2k$ from both sides of the equation: $14k - 2k - 52 = 2k - 2k + 20$ , which simplifies to $12k - 52 = 20$ .
Find Initial Numbers: Adding $52$ to both sides to isolate the term with $k$ , we get $12k - 52 + 52 = 20 + 52$ , which simplifies to $12k = 72$ .
Calculate Total People: Dividing both sides by $12$ to solve for $k$ , we get $k = \frac{72}{12}$ , which simplifies to $k = 6$ .
Calculate Total People: Dividing both sides by $12$ to solve for $k$ , we get $k = \frac{72}{12}$ , which simplifies to $k = 6$ .Now that we have the value of $k$ , we can find the initial number of children and adults. The initial number of children is $C = 7k = 7 \times 6 = 42$ , and the initial number of adults is $A = 2k = 2 \times 6 = 12$ .
Calculate Total People: Dividing both sides by $12$ to solve for $k$ , we get $k = \frac{72}{12}$ , which simplifies to $k = 6$ .Now that we have the value of $k$ , we can find the initial number of children and adults. The initial number of children is $C = 7k = 7 \times 6 = 42$ , and the initial number of adults is $A = 2k = 2 \times 6 = 12$ .The total number of people at the exhibition at first is the sum of the initial number of children and adults, which is $42 + 12 = 54$ .