$3$ . Tom and Rob are brothers who like to make bets about the outcomes of different contests between them. Before the last bet, the ratio of the amount of Tom's money to the amount of Rob's money was $4: 7$ . Rob lost the latest competition, and now the ratio of the amount of Tom's money to the amount of Rob's money is $8: 3$ . If Rob had $\$ 280$ before the last competition, how much does Rob have now that he lost the bet?

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3

. Tom and Rob are brothers who like to make bets about the outcomes of different contests between them. Before the last bet, the ratio of the amount of Tom's money to the amount of Rob's money was

4: 7

. Rob lost the latest competition, and now the ratio of the amount of Tom's money to the amount of Rob's money is

8: 3

. If Rob had

\$ 280

before the last competition, how much does Rob have now that he lost the bet?

Set Up Ratio Equation: Let's call Tom's original amount of money $T$ and Rob's original amount of money $R$ . The original ratio is $4:7$ , so we can write it as $\frac{T}{R} = \frac{4}{7}$ .
Find Tom's Original Amount: We know Rob had $\$280$ before the last competition, so $R = 280$ .
Calculate Tom's Amount: Using the original ratio, we can find Tom's original amount of money: $T = \left(\frac{4}{7}\right) \times R = \left(\frac{4}{7}\right) \times 280$ .
Determine New Ratio: Calculating Tom's original amount: $T = \left(\frac{4}{7}\right) \times 280 = 4 \times 40 = 160$ . So, Tom had $\$160$ before the last competition.
Express Tom's New Amount: Now, the new ratio after Rob lost the bet is $8:3$ , which means $T/R = 8/3$ .
Substitute in New Ratio: Let's call Rob's new amount of money $R_{\text{new}}$ . We can express Tom's new amount of money as $T_{\text{new}} = T + (R - R_{\text{new}})$ because Tom wins what Rob loses.
Solve for $R_{\text{new}}$ : Using the new ratio, we can write $\frac{T_{\text{new}}}{R_{\text{new}}} = \frac{8}{3}$ . Substituting $T_{\text{new}}$ with $T + (R - R_{\text{new}})$ , we get $\frac{T + (R - R_{\text{new}})}{R_{\text{new}}} = \frac{8}{3}$ .
Expand and Simplify: Substitute $T$ with $160$ and $R$ with $280$ : $\left(160 + \left(280 - R_{\text{new}}\right)\right)/R_{\text{new}} = \frac{8}{3}$ .
Combine Like Terms: Cross-multiply to solve for $R_{\text{new}}$ : $3\times(160 + (280 - R_{\text{new}})) = 8\times R_{\text{new}}$ .
Divide to Find $R_{\text{new}}$ : Expand and simplify: $480 + 840 - 3\cdot R_{\text{new}} = 8\cdot R_{\text{new}}$ .
Calculate Rob's New Amount: Combine like terms: $1320 = 11 \times R_{\text{new}}$ .
Calculate Rob's New Amount: Combine like terms: $1320 = 11 \times R_{\text{new}}$ .Divide both sides by $11$ to find $R_{\text{new}}$ : $R_{\text{new}} = \frac{1320}{11}$ .
Calculate Rob's New Amount: Combine like terms: $1320 = 11 \times R_{\text{new}}$ . Divide both sides by $11$ to find $R_{\text{new}}$ : $R_{\text{new}} = \frac{1320}{11}$ . Calculate $R_{\text{new}}$ : $R_{\text{new}} = 120$ . So, Rob has $\$120$ now after losing the bet.