Valeria and Christina both want to run for class president. In order to appear on the ballot, they must collect a certain number of signatures. So far, Valeria has $26$ signatures and Christina has $53$ . Now, Valeria is collecting signatures at a rate of $6$ per day, whereas Christina is collecting $3$ signatures per day. Assuming that their rates stay the same, the two will eventually collect the same number of signatures. How many signatures will they both have at that time? How many days will it take? Valeria and Christina will each have collected signatures. It will take days.

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Solve proportions: word problems

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Q. Valeria and Christina both want to run for class president. In order to appear on the ballot, they must collect a certain number of signatures. So far, Valeria has

26

signatures and Christina has

53

. Now, Valeria is collecting signatures at a rate of

6

per day, whereas Christina is collecting

3

signatures per day. Assuming that their rates stay the same, the two will eventually collect the same number of signatures. How many signatures will they both have at that time? How many days will it take? Valeria and Christina will each have collected ____ signatures. It will take ____ days.

Set up equations: Let's set up the equations for Valeria and Christina's signature collections. Valeria starts with $26$ signatures and collects $6$ more per day. Christina starts with $53$ signatures and collects $3$ more per day. We need to find when they have the same number of signatures.
Equations for signatures: Set up the equations for each: $\newline$ Valeria's signatures = $26 + 6d$ $\newline$ Christina's signatures = $53 + 3d$ $\newline$ where $d$ is the number of days after which they have the same number of signatures.
Set equations equal: To find when they have the same number of signatures, set the equations equal to each other: $\newline$ $26 + 6d = 53 + 3d$
Solve for d: Solve for d: $\newline$ $6d - 3d = 53 - 26$ $\newline$ $3d = 27$ $\newline$ $d = \frac{27}{3}$ $\newline$ $d = 9$
Calculate signatures after $9$ days: Now, calculate the number of signatures each will have after $9$ days: $\newline$ Valeria's signatures = $26 + 6 \times 9 = 26 + 54 = 80$ $\newline$ Christina's signatures = $53 + 3 \times 9 = 53 + 27 = 80$