Question 24·Hard·Ratios, Rates, Proportional Relationships, and Units
A construction crew must prepare 5,600 kilograms of gravel aggregate that is 32% sand by mass.
They can draw material from two stockpiles.
- Stockpile P: 20% sand, the rest stone.
- Stockpile Q: 40% sand, the rest stone.
Two conveyor belts deliver the material into a mixer. During operation, the belt that carries material from Stockpile Q delivers gravel 1.5 times as fast (in kilograms per minute) as the belt that carries material from Stockpile P. The belts are turned on at the same moment and turned off at the same moment when the mixer reaches 5,600 kilograms, so the masses of gravel added from the two stockpiles are in the same ratio as the delivery rates.
How many kilograms of material from Stockpile Q are in the final 5,600-kilogram mixture?
(Express the answer as an integer)
For mixture problems that also involve rates, first translate the rate information into a ratio of amounts: if one source runs 1.5 times as fast for the same time, then its mass is 1.5 times the other's. Let one amount be a variable (for example, from P, from Q), and immediately use the given total mass to write a simple equation like . Solve for the variable in one or two steps, then use the ratio to get the other amount. Percent compositions (like 20%, 40%, 32%) are often there to check your work, but they are not always needed to find the requested quantity, so focus first on the quickest relationship that directly uses the total and the rate ratio.
Hints
Connect belt speed to mass delivered
If one belt delivers material 1.5 times as fast as the other, and they run for the same amount of time, how do the masses delivered from the two stockpiles compare?
Write an equation for the total mass
Let the mass from Stockpile P be kilograms. Using the speed relationship, express the mass from Stockpile Q in terms of , and then write an equation saying that the sum of the masses from P and Q is 5,600 kilograms.
Solve for the mass from Q
Once you solve for , multiply by the appropriate factor to get the mass from Q. Afterward, you can use the 20% and 40% sand percentages to check that the mixture really is 32% sand, but you do not need them to set up the main equation.
Desmos Guide
Solve for the mass from Stockpile P
In Desmos, type the equation 0.2x + 0.4(1.5x) = 0.32*5600 (this uses for the mass from P and for the mass from Q). Desmos will show the value of that satisfies the equation; this value is the mass from Stockpile P.
Find the mass from Stockpile Q
On a new line, type 1.5x using the value you just found. The resulting value of this expression is the mass from Stockpile Q in kilograms.
Step-by-step Explanation
Relate the amounts from P and Q using the belt speeds
Let be the number of kilograms taken from Stockpile P.
The belt from Stockpile Q delivers material 1.5 times as fast as the belt from P, and both belts run for the same amount of time. That means the mass from Q is 1.5 times the mass from P, so
where is the number of kilograms taken from Stockpile Q.
Use the total mass to find the amount from P
The final mixture must have a total mass of 5,600 kilograms:
Substitute into this equation:
Now solve for :
So 2,240 kilograms come from Stockpile P.
Find the amount from Q and (optionally) check the sand percentage
Use and to find the mass from Stockpile Q:
(Quick check using the sand percentages: sand from P is kg, sand from Q is kg, for a total of kg of sand. Then , or 32% sand, which matches the requirement.)
Therefore, the final 5,600-kilogram mixture contains 3360 kilograms from Stockpile Q.