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?

Question

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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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 $.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 $.

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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