There are ten times as many pennies as nickels, five more dimes than nickels, two fewer quarters than dimes, and twice as many silver dollars as quarters. Express the number of each coin in terms of the variable if there are n nickels. Exercise

n nickels p = 10n d = n + 5 q = n + 3 s = 2n + 6 Exercise

There are ten times as many pennies as nickels, five more dimes than nickels, two fewer quarters than dimes, and twice as many silver dollars as quarters. Express the number of each coin in terms of the variable if there are p pennies. Exercise

p pennies n = 0.1p d = 0.1p + 5 q = 0.1p + 3 s = 0.2p + 6 Exercise

There are ten times as many pennies as nickels, five more dimes than nickels, two fewer quarters than dimes, and twice as many silver dollars as quarters. Express the number of each coin in terms of the variable if there are d dimes. Exercise

d dimes p = 10d – 50 n = d – 5 q = d – 2 s = 2d – 4 Exercise

There are ten times as many pennies as nickels, five more dimes than nickels, two fewer quarters than dimes, and twice as many silver dollars as quarters. Express the number of each coin in terms of the variable if there are q quarters. Exercise

q quarters p = 10q – 30 n = q – 3 d = q + 2 s = 2q Exercise

If there are ten nickels, what is the total value of the coins? $32.25 Exercise

Steps to Solving Problems with Equations 1.Represent an unknown quantity with a variable. 2.When necessary, represent other conditions in the problem in terms of the same variable.

Steps to Solving Problems with Equations 3. Identify two equal quantities in the problem. 4. Write and solve an equation.

Choose a variable for the unknown in the following problems. Then write an equation that could be used to solve the problem. Do not solve. Example 1

Coach Mahoney ran six times as far as he swam and cycled thirty times as far as he swam. If he covered a total of 37 mi. in the triathlon, how far did he swim? Let x = length of swim. 6x = length of run 30x = length of bicycle ride

Let x = length of swim. 6x = length of run 30x = length of bicycle ride length of swim + length of run + length of bicycle ride = 37 x + 6x + 30x = 37

Choose a variable for the unknown in the following problems. Then write an equation that could be used to solve the problem. Do not solve. Example 2

A soccer field has a perimeter of 380 yd., and it’s length is 40 yd. longer than its width. Find the dimensions of the field. Let w = width w + 40 = length

Let w = width. w + 40 = length width + 2 length = perimeter 2w + 2 (w + ) =

A carpenter is nailing baseboard molding along a wall that is 88 in. long. He has nailed in place a piece of molding that is 32 in. long. What length of molding does he need to finish the job? Example 3

Let m = length of molding needed. m + 32 = molding needed + molding nailed = length of baseboard

m + 32 = m + 32 – 32 = 88 – m = 55 in =

The drama club performed the school play on Friday and Saturday nights. Attendance at the Saturday performance was 1 times the attendance on Friday Example 4

If there were 95 people in attendance on Saturday, what was the total attendance for both performances? Let f = Friday’s attendance Friday’s attendance = Saturday’s attendance 1.25f = 95

171 people for the two nights 1.25f = f = = 171

When three-fourths of a number is added to 12, the result is 18. What is the number? n + 12 = 18 n = 8 Example

Twice a number divided by 15 is seven-fifths. 2n 15 = = n = 21 2 Example

A 2 by 4 is really 1.5 in. by 3.5 in. How many, if laid flat, would be needed to make a deck 8 ft. wide? 3.5x = 12(8) x = 27.4, so 28 will be needed Example

If it takes 3 c. of flour to make two dozen cookies, how many cookies can be made with 8 c. of flour? 3 24 = = 8x8x 8x8x x = 64 cookies Example

If a recipe calls for 3 c. flour and 1 c. sugar for two dozen cookies, how many cookies can be made if 8 c. flour and 5 c. sugar are available? Example

for the flour: x = 64 cookies 24 = = 5x5x 5x5x for the sugar: x = 72 The limiting factor is flour; only 64 cookies can be made.

Two additional questions: How much sugar is left over? How much additional flour is needed to make 72 cookies? c. left over 1 c.

If 1 c. corn syrup is equivalent in sweetness to c. sugar, how much corn syrup is needed to replace the sugar if the recipe calls for 2 c. sugar? Example

= = n n n = 3 c. corn syrup

How many pounds of raisins at $2.25/lb. should be mixed with peanuts at $3.50/lb. to make a 50 lb. batch of trail mix that costs $3.00/lb.? x = raisins; 50 – x = peanuts 2.25x + 3.5(50 – x) = 3(50) x = 20 lb. of raisins Example

How many pounds of a $12.50/lb. premium blend of coffee should be mixed with a $4.50/lb. cheap blend to make the house blend, which sells for $15 for a 2 lb. bag? x = premium 12.5x + 4.5(2 – x) = 15 x = 0.75 lb. Example

What was the average speed of a motorist who went 50 mi./hr. for 1.5 hr. and then 40 mi./hr. for 45 min.? 50(1.5) + 40(0.75) = r (2.25) r = 46.6 mi./hr. Example

If Bill needs to make a 500 mi. trip in 12 hr. but gets stuck in a traffic jam for half an hour, what must his average speed be during the remaining time to still make it to his destination on time? 500 = r (11.5) r ≈ mi./hr. Example