# You have a stock of two blends of dry mix sand and cement that you wish to blend to give you 10 tons at 40% cement content. ?

## One of these mixes has 20% cement content and the other has 70% cement content. How many tons of each need to be mixed together to give you the target concentration and weight.

Oct 23, 2017

6 tons at 20% blend
4 tons at 70% blend

#### Explanation:

Target is a blend with 40% cement

Let the amount of 20% material be designated as ${M}_{20}$
Let the amount of 70% material be designated as ${M}_{70}$

${M}_{20} + {M}_{70} = 10 \text{ } \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . . E q u a t i o n \left(1\right)$

20%M_20+70%M_70 = 40%10" "............Equation(2)

From $E q n \left(1\right) \text{ "M_20=10-M_70" } \ldots \ldots \ldots \ldots E q u a t i o n \left({1}_{a}\right)$

Using $E q n \left({1}_{a}\right)$ substitute for ${M}_{20}$ in $E q n \left(2\right)$

color(white)("dddddddddd")color(green)(20/100(color(red)(M_20))+70/100M_70=40/100xx10

$\textcolor{w h i t e}{\text{dddd")color(white)("dd}} \frac{20}{100} \left(\textcolor{red}{10 - {M}_{70}}\right) + \frac{70}{100} {M}_{70} = \frac{400}{100}$

Multiply both sides by 100 to get rid of the fraction

$\textcolor{w h i t e}{\text{ddddddd")color(white)("dd}} 20 \left(10 - {M}_{70}\right) + 70 {M}_{70} = 400$

$\textcolor{w h i t e}{\text{ddddddddddddd")200color(white)("dddd}} + 50 {M}_{70} = 400$

$\textcolor{w h i t e}{\text{dddddddddddddddddddddddd")color(blue)(M_70=(400-200)/50 = 4" tons}}$
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Thus $\textcolor{b l u e}{{M}_{20} = 10 - 4 = 6 \text{ tons}}$

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color(brown)("Checking LHS - RHS is "40%xx10=4)

$\left(\frac{20}{100} \times 6\right) + \left(\frac{70}{100} \times 4\right) = 4$

LHS=RHS thus proven to be true

color(white)("d")