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Emily had some red beads and blue beads in the ratio
5:2. She had 54 more red beads than blue beads. When she gave away 18 red beads, what was the new ratio of the number of red beads to that of the blue beads?

Emily had some red beads and blue beads in the ratio $5: 2$ . She had $54$ more red beads than blue beads. When she gave away $18$ red beads, what was the new ratio of the number of red beads to that of the blue beads?

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Solve proportions: word problems

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Q. Emily had some red beads and blue beads in the ratio

5: 2

. She had

54

more red beads than blue beads. When she gave away

18

red beads, what was the new ratio of the number of red beads to that of the blue beads?

Define Bead Numbers: Let's define the number of red beads as $R$ and the number of blue beads as $B$ . Given the ratio $R:B = 5:2$ and $R = B + 54$ .
Express in Terms of x: Using the ratio, we can express $R$ and $B$ in terms of a common variable $x$ , where $R = 5x$ and $B = 2x$ .
Solve for x: Substituting $R = 5x$ into the equation $R = B + 54$ , we get $5x = 2x + 54$ . Solving for $x$ , we subtract $2x$ from both sides to get $3x = 54$ . Then, $x = 18$ .
Substitute x Values: Now, substituting $x = 18$ back into the expressions for $R$ and $B$ , we find $R = 5 \times 18 = 90$ and $B = 2 \times 18 = 36$ .
Update Red Bead Count: Emily gave away $18$ red beads, so the new number of red beads is $90 - 18 = 72$ .
Simplify New Ratio: The new ratio of red beads to blue beads is $72:36$ . Simplifying this ratio by dividing both numbers by $36$ , we get $2:1$ .

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