Question

1. celia and ryan are starting a nutrition program. celia currently consumes 1200 calories a day and will increase that number by 100 calories each day. ryan currently consumes 3230 calories a day and will decrease that number by 190 each day. they will continue this pattern until they are both consuming the same number of calories per day. in how many days will that be? 4. aaron needs to take out a loan to purchase a motorcycle. at one bank, he would pay $2500 initially and $150 each month for the loan. at another bank, he would pay $3000 initially and $125 each month. after how many months will the two loan payments be the same?

Explanation:

Step1: Set up equations for Celia and Ryan

Let $d$ be the number of days. Celia's calorie - consumption is given by the equation $C = 1200+100d$. Ryan's calorie - consumption is given by the equation $R = 3230 - 190d$. We want to find when $C = R$, so we set up the equation $1200+100d=3230 - 190d$.

Step2: Combine like terms

Add $190d$ to both sides of the equation: $1200+100d + 190d=3230-190d + 190d$, which simplifies to $1200 + 290d=3230$. Then subtract $1200$ from both sides: $1200-1200 + 290d=3230 - 1200$, resulting in $290d=2030$.

Step3: Solve for $d$

Divide both sides of the equation $290d=2030$ by $290$: $d=\frac{2030}{290}=7$.

Step4: Set up equations for the loan payments

Let $m$ be the number of months. For the first bank, the loan payment is $P_1 = 2500+150m$. For the second bank, the loan payment is $P_2 = 3000+125m$. We want to find when $P_1 = P_2$, so we set up the equation $2500+150m=3000+125m$.

Step5: Combine like terms

Subtract $125m$ from both sides: $2500+150m-125m=3000+125m-125m$, which gives $2500 + 25m=3000$. Then subtract $2500$ from both sides: $2500-2500+25m=3000 - 2500$, resulting in $25m=500$.

Step6: Solve for $m$

Divide both sides of the equation $25m=500$ by $25$: $m=\frac{500}{25}=20$.

Answer:

For Celia and Ryan, it will be 7 days.
For Aaron's loan payments, it will be 20 months.