Mia has a jar containing nickels and dimes worth $ 4.8 \$4.8 in total. If she has 12 12 more dimes than nickels, how many of each coin does she have?

N i c k e l s = 24 ; D i m e s = 36 Nickels=24;Dimes=36 N i c k e l s = 24 ; D im es = 36

N i c k e l s = 36 ; D i m e s = 24 Nickels=36;Dimes=24

N i c k e l s = 30 ; D i m e s = 42 Nickels=30;Dimes=42 N i c k e l s = 30 ; D im es = 42

N i c k e l s = 42 ; D i m e s = 30 Nickels=42;Dimes=30 N i c k e l s = 42 ; D im es = 30

A technician needs to prepare a disinfectant by mixing a 70 % 70\% isopropyl alcohol solution with some 50 % 50\% solution to obtain 65 % 65\% alcohol. If the technician uses 500 m L ⁡ 500\operatorname{\mathrm{mL}} of the 50 % 50\% solution, how many m L mL of the 70 % 70\% solution must be added?

1500 mL ⁡ 1500\operatorname{\mathrm{m}\mathrm{L}} 1500 mL

600 mL ⁡ 600\operatorname{\mathrm{m}\mathrm{L}} 600 mL

667 mL ⁡ 667\operatorname{\mathrm{m}\mathrm{L}} 667 mL

67 mL ⁡ 67\operatorname{\mathrm{m}\mathrm{L}} 67 mL

Elena has $18,500 to invest. She invests some of it at 8.4 % 8.4\% annual simple interest for 1 1 year, and the remainder at 11.6 % 11.6\% annual simple interest for 8 8 months. At the end of the year, her total interest earned is $1,500. How much did she invest at each rate?

$10,399.92 at 8.4 % 8.4\% and $8100.08 at 11.6 % 11.6\%

$11,000 at 8.4 % 8.4\% and $7,500 at 11.6 % 11.6\%

$7,500 at 8.4 % 8.4\% and $11,000 at 11.6 % 11.6\%

$8100.08 at 8.4 % 8.4\% and $10,399.92 at 11.6 % 11.6\%

How do you solve mixture problems involving coins like dimes and nickels?

To solve mixture problems with coins such as dimes and nickels, start by defining variables for the number of each coin type. For example, let d be the number of dimes and n be the number of nickels. Use the total value to build an equation: 0.10 × d + 0.05 × n = 2.20 . If given a relationship like "eight more nickels than dimes," express one variable in terms of the other, e.g., n = d + 8 . Substitute this into the value equation to have one variable, then solve algebraically. This method helps find the exact number of each coin type in the mixture.

3. Solving Word Problems

Mixture Problem Solving

3. Solving Word Problems

Mixture Problem Solving: Videos & Practice Problems

Video Lessons

Topic summary

Mixture problems involve combining quantities, such as coins or solutions, to form a total amount. By expressing parts as products and summing them, we build equations like $total = d × 0.1 + n × 0.05$ or $0.5 × 14 = 0.4 × x + 0.7 × y$ . Using substitution, variables are rewritten in terms of one another to solve for unknowns. When percents are involved, convert them to decimals for calculations. This approach applies to various contexts, reinforcing skills in algebraic manipulation and problem-solving with polynomials and terms.

concept

Intro to Mixture Problems

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Intro to Mixture Problems Video Summary

Mixture problems involve combining two or more quantities to form a single mixture, and they can be solved using systematic problem-solving steps similar to other word problems. These problems often appear in various contexts such as money and coins, tile laying, ticket sales, or mixing chemical solutions. A common example is determining the number of different coins in a collection when given the total value and a relationship between the quantities.

Consider a scenario where the total amount of money is \$2.20, composed of dimes and nickels. If there are eight more nickels than dimes, we can represent the number of dimes as d and the number of nickels as n. The total value equation is constructed by multiplying the number of each coin by its value and summing these amounts:

\[0.10d + 0.05n = 2.20\]

Since the number of nickels is eight more than the number of dimes, we express this relationship as:

\[n = d + 8\]

Substituting this into the total value equation gives:

\[0.10d + 0.05(d + 8) = 2.20\]

Distributing and simplifying leads to:

\[0.10d + 0.05d + 0.40 = 2.20\]\[0.15d + 0.40 = 2.20\]

Isolating the variable term by subtracting 0.40 from both sides:

\[0.15d = 1.80\]

Dividing both sides by 0.15 to solve for d:

\[d = \frac{1.80}{0.15} = 12\]

Knowing the number of dimes, the number of nickels is:

\[n = 12 + 8 = 20\]

This approach highlights the key steps in solving mixture problems: first, identify the total quantity and the individual components; second, translate the problem into an algebraic equation by expressing one variable in terms of another; third, substitute and simplify to solve for a single variable; and finally, interpret the solution in the context of the problem.

Understanding how to set up and solve these equations is essential for tackling a wide range of mixture problems, whether involving money, materials, or chemical solutions. Mastery of this method enhances problem-solving skills and builds a strong foundation for more complex applications.

Problem

Mia has a jar containing nickels and dimes worth $dollar sign 4.8$ in total. If she has $12$ more dimes than nickels, how many of each coin does she have?

$N i c k e l s = 36; D i m e s = 24$

$Nickels=24;Dimes=36$

$Nickels=30;Dimes=42$

$Nickels=42;Dimes=30$

example

Intro to Mixture Problems Example 1

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Intro to Mixture Problems Example 1 Video Summary

To determine the number of student and adult tickets Market purchased for a basketball game, we start by defining variables and setting up equations based on the given information. Student tickets cost \$12 each, adult tickets cost \$20 each, and Market bought a total of 28 tickets, spending \$464 in total.

Let s represent the number of student tickets and a represent the number of adult tickets. The total cost equation can be expressed as:

\[12s + 20a = 464\]

Additionally, since the total number of tickets is 28, we have the relationship:

\[s + a = 28\]

Solving for s in terms of a gives:

\[s = 28 - a\]

Substituting this expression for s into the total cost equation yields:

\[12(28 - a) + 20a = 464\]

Distributing 12 across the terms inside the parentheses results in:

\[336 - 12a + 20a = 464\]

Combining like terms simplifies the equation to:

\[336 + 8a = 464\]

Subtracting 336 from both sides gives:

\[8a = 128\]

Dividing both sides by 8 isolates a:

\[a = 16\]

Using the value of a to find s:

\[s = 28 - 16 = 12\]

Therefore, Market purchased 12 student tickets and 16 adult tickets. This problem demonstrates how to apply systems of linear equations to real-world scenarios, using substitution to solve for unknown variables. Understanding how to translate word problems into algebraic expressions and manipulate equations is essential for solving similar problems involving cost, quantity, and total values.

Problem

Maya needs $48$ sq ft of tile for a backsplash. Basic tiles cost \$9 per sq ft and designer tiles cost \$25 per sq ft. She wants the overall average cost to be $dollar sign 14$ per sq ft. How many square feet of each tile should she use?

Basic tiles $equals 18 sq of ft of$ ; Designer tiles $equals 30 sq of ft of$

Basic tiles $equals 30 sq of ft of$ ; designer tiles $equals 18 sq of ft of$

Basic tiles $equals 15 sq of ft of$ ; Designer tiles $equals 33 sq of ft of$

Basic tiles $equals 33 sq of ft of$ ; Designer tiles $equals 15 sq of ft of$

concept

Mixture Problems Involving Percents

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Mixture Problems Involving Percents Video Summary

Mixture problems involving percents require applying the same fundamental methods used in standard mixture problems, with the added step of converting percentages to decimals. When solving these problems, the key is to express each component of the mixture as a product of its quantity and concentration, then sum these products to equal the total desired concentration.

For example, consider a chemist who needs 14 liters of a 50% acid solution but only has access to 40% and 70% acid solutions. To determine how much of each solution to mix, define variables for the unknown quantities: let x represent liters of the 40% solution and y represent liters of the 70% solution. The total volume equation is:

\[x + y = 14\]

Next, express the acid content from each solution as the product of the volume and its decimal concentration (percent converted to decimal). The total acid content in the final mixture is 50% of 14 liters, or:

\[0.4x + 0.7y = 0.5 \times 14\]

Substituting the total acid content:

\[0.4x + 0.7y = 7\]

Using the total volume equation, solve for one variable, for example:

\[y = 14 - x\]

Substitute this into the acid content equation:

\[0.4x + 0.7(14 - x) = 7\]

Distribute and simplify:

\[0.4x + 9.8 - 0.7x = 7\]\[-0.3x + 9.8 = 7\]

Isolate x:

\[-0.3x = 7 - 9.8\]\[-0.3x = -2.8\]\[x = \frac{-2.8}{-0.3} = 9.\overline{3}\]

This means approximately 9.33 liters of the 40% acid solution are needed. To find y, substitute back:

\[y = 14 - 9.33 = 4.67\]

Thus, 4.67 liters of the 70% acid solution are required. This approach highlights that solving percent mixture problems involves setting up equations based on total volume and total concentration, converting percentages to decimals, and solving for unknown quantities. Understanding how to translate percent concentrations into decimal form and apply algebraic methods is essential for accurately solving these types of problems.

Problem

A technician needs to prepare a disinfectant by mixing a $70 percent sign$ isopropyl alcohol solution with some $50 percent sign$ solution to obtain $65 percent sign$ alcohol. If the technician uses $500 m L$ of the $50 percent sign$ solution, how many $m L$ of the $70 percent sign$ solution must be added?

$1500\operatorname{\mathrm{m}\mathrm{L}}$

$600\operatorname{\mathrm{m}\mathrm{L}}$

$667\operatorname{\mathrm{m}\mathrm{L}}$

$67\operatorname{\mathrm{m}\mathrm{L}}$

Problem

A candy maker has $300 mL$ of a $45 percent sign$ sugar syrup. She wants to dilute it with pure water to make a $30 percent sign$ syrup. How many $m L$ of water should she add?

$450 mL$

$750 mL$

$150\operatorname{\mathrm{m}\mathrm{L}}$

$250\operatorname{\mathrm{m}\mathrm{L}}$

example

Example 2: Mixing Simple Interest

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Example 2: Mixing Simple Interest Video Summary

When solving problems involving investments with different interest rates, the simple interest formula is essential. Simple interest is calculated as the product of the principal amount, the interest rate, and the time period. In this scenario, a total of \$15,000 is split between two investments: one at 7% annual simple interest and the other at 9% annual simple interest. The total interest earned in one year from both investments is \$1,112.

To find the amounts invested at each rate, let x represent the amount invested at 7%, and y represent the amount invested at 9%. Since the total investment is \$15,000, the relationship between x and y is:

\[x + y = 15,000\]

The total interest earned from both investments can be expressed using the simple interest formula for each part:

\[0.07x + 0.09y = 1,112\]

To solve for one variable, express y in terms of x:

\[y = 15,000 - x\]

Substitute this into the interest equation:

\[0.07x + 0.09(15,000 - x) = 1,112\]

Distribute the 0.09:

\[0.07x + 1,350 - 0.09x = 1,112\]

Combine like terms:

\[-0.02x + 1,350 = 1,112\]

Subtract 1,350 from both sides:

\[-0.02x = 1,112 - 1,350\]\[-0.02x = -238\]

Divide both sides by -0.02 to isolate x:

\[x = \frac{-238}{-0.02} = 11,900\]

This means \$11,900 was invested at 7%. To find the amount invested at 9%, substitute back into the equation for y:

\[y = 15,000 - 11,900 = 3,100\]

Therefore, \$3,100 was invested at 9%. This approach demonstrates how to apply the simple interest formula and solve systems of linear equations to determine investment distributions based on total interest earned.

Problem

A retirement fund invests some money in government bonds at $2.5 percent sign$ annual simple interest, and \$5,000 more than that amount in corporate bonds at $4 percent sign$ . If the total annual interest earned is \$1,450, how much was invested in each type of bond?

Government bonds = \$24,230.77; Corporate bonds = \$19,230.77

Government bonds = \$19,230.77; Corporate bonds = \$24,230.77

Government bonds = \$18,500.50; Corporate bonds = \$23,500.50

Government bonds = \$23,500.50; Corporate bonds = \$18,500.50

Problem

Elena has \$18,500 to invest. She invests some of it at $8.4 percent sign$ annual simple interest for $1$ year, and the remainder at $11.6 percent sign$ annual simple interest for $8$ months. At the end of the year, her total interest earned is \$1,500. How much did she invest at each rate?

\$11,000 at $8.4 percent sign$ and \$7,500 at $11.6 percent sign$

\$7,500 at $8.4 percent sign$ and \$11,000 at $11.6 percent sign$

\$10,399.92 at $8.4 percent sign$ and \$8100.08 at $11.6 percent sign$

\$8100.08 at $8.4 percent sign$ and \$10,399.92 at $11.6 percent sign$

Here’s what students ask on this topic:

To solve mixture problems with coins such as dimes and nickels, start by defining variables for the number of each coin type. For example, let $d$ be the number of dimes and $n$ be the number of nickels. Use the total value to build an equation: $0.10 \times d + 0.05 \times n = 2.20$ . If given a relationship like "eight more nickels than dimes," express one variable in terms of the other, e.g., $n = d + 8$ . Substitute this into the value equation to have one variable, then solve algebraically. This method helps find the exact number of each coin type in the mixture.

When solving mixture problems with percentages, first identify the quantities and their respective concentrations. Assign variables to unknown amounts, such as $x$ and $y$ . Convert percentages to decimals (e.g., 40% as 0.4). Build an equation representing the total amount of the mixture, like $x + y = 14$ . Then, write an equation for the total amount of the substance of interest (acid, for example): $0.4 \times x + 0.7 \times y = 0.5 \times 14$ . Substitute one variable from the first equation into the second, solve for one variable, then find the other. This approach ensures accurate mixing ratios.

Expressing one variable in terms of another is essential for solving mixture problems with multiple unknowns. For example, if you know there are eight more nickels than dimes, and you let $d$ represent dimes, then nickels $n$ can be written as $n = d + 8$ . This substitution reduces the number of variables in your equation, allowing you to solve for a single variable. After finding the value of one variable, substitute back to find the other. This technique simplifies complex problems into manageable algebraic equations.

Converting percentages to decimals in mixture problems is crucial because it allows you to perform accurate multiplication when calculating parts of a mixture. For example, 50% is converted to 0.5 to represent the fraction of the total amount. This conversion enables you to write equations like $0.5 \times 14$ to find the amount of a substance in a solution. Using decimals ensures consistency in calculations and avoids errors that can arise from working directly with percentages.

To build an equation for the total amount in a mixture problem, identify each component's quantity and its contribution to the total. For example, if mixing solutions with different concentrations, multiply each quantity by its concentration (expressed as a decimal). Sum these products to equal the total amount of the desired mixture. Mathematically, this looks like $x \times 0.4 + y \times 0.7 = 14 \times 0.5$ . This equation represents the total amount of the substance of interest in the mixture and is the foundation for solving the problem.

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Patrick Ford

Physics and Math Lead Instructor

Mixture Problem Solving: Videos & Practice Problems

Intro to Mixture Problems

Intro to Mixture Problems Video Summary

Mia has a jar containing nickels and dimes worth $dollar sign 4.8$ in total. If she has $12$ more dimes than nickels, how many of each coin does she have?

Intro to Mixture Problems Example 1

Intro to Mixture Problems Example 1 Video Summary

Maya needs $48$ sq ft of tile for a backsplash. Basic tiles cost \$9 per sq ft and designer tiles cost \$25 per sq ft. She wants the overall average cost to be $dollar sign 14$ per sq ft. How many square feet of each tile should she use?

Mixture Problems Involving Percents

Mixture Problems Involving Percents Video Summary

A candy maker has $300 mL$ of a $45 percent sign$ sugar syrup. She wants to dilute it with pure water to make a $30 percent sign$ syrup. How many $m L$ of water should she add?

Example 2: Mixing Simple Interest

Example 2: Mixing Simple Interest Video Summary

A retirement fund invests some money in government bonds at $2.5 percent sign$ annual simple interest, and \$5,000 more than that amount in corporate bonds at $4 percent sign$ . If the total annual interest earned is \$1,450, how much was invested in each type of bond?

Elena has \$18,500 to invest. She invests some of it at $8.4 percent sign$ annual simple interest for $1$ year, and the remainder at $11.6 percent sign$ annual simple interest for $8$ months. At the end of the year, her total interest earned is \$1,500. How much did she invest at each rate?

Here’s what students ask on this topic:

How do you solve mixture problems involving coins like dimes and nickels?

What is the step-by-step method to solve mixture problems involving percentages?

How do you express one variable in terms of another in mixture problems?

Why do you convert percentages to decimals in mixture problems?

How do you build an equation for the total amount in a mixture problem?

Your Intermediate Algebra tutors

Mixture Problem Solving: Videos & Practice Problems

Intro to Mixture Problems

Intro to Mixture Problems Video Summary

Mia has a jar containing nickels and dimes worth $4.8\$4.8 in total. If she has 1212 more dimes than nickels, how many of each coin does she have?

Intro to Mixture Problems Example 1

Intro to Mixture Problems Example 1 Video Summary

Maya needs 4848 sq ft of tile for a backsplash. Basic tiles cost \$9 per sq ft and designer tiles cost \$25 per sq ft. She wants the overall average cost to be $14\$14 per sq ft. How many square feet of each tile should she use?

Mixture Problems Involving Percents

Mixture Problems Involving Percents Video Summary

A technician needs to prepare a disinfectant by mixing a 70%70\% isopropyl alcohol solution with some 50%50\% solution to obtain 65%65\% alcohol. If the technician uses 500mL500\operatorname{\mathrm{mL}} of the 50%50\% solution, how many mLmL of the 70%70\% solution must be added?

A candy maker has 300mL300\operatorname{mL} of a 45%45\% sugar syrup. She wants to dilute it with pure water to make a 30%30\% syrup. How many mLmL of water should she add?

Example 2: Mixing Simple Interest

Example 2: Mixing Simple Interest Video Summary

A retirement fund invests some money in government bonds at 2.5%2.5\% annual simple interest, and \$5,000 more than that amount in corporate bonds at 4%4\%. If the total annual interest earned is \$1,450, how much was invested in each type of bond?

Elena has \$18,500 to invest. She invests some of it at 8.4%8.4\% annual simple interest for 11 year, and the remainder at 11.6%11.6\% annual simple interest for 88 months. At the end of the year, her total interest earned is \$1,500. How much did she invest at each rate?

Here’s what students ask on this topic:

How do you solve mixture problems involving coins like dimes and nickels?

How do you solve mixture problems involving coins like dimes and nickels?

What is the step-by-step method to solve mixture problems involving percentages?

What is the step-by-step method to solve mixture problems involving percentages?

How do you express one variable in terms of another in mixture problems?

How do you express one variable in terms of another in mixture problems?

Why do you convert percentages to decimals in mixture problems?

Why do you convert percentages to decimals in mixture problems?

How do you build an equation for the total amount in a mixture problem?

How do you build an equation for the total amount in a mixture problem?

Your Intermediate Algebra tutors

Mia has a jar containing nickels and dimes worth $dollar sign 4.8$ in total. If she has $12$ more dimes than nickels, how many of each coin does she have?

Maya needs $48$ sq ft of tile for a backsplash. Basic tiles cost \$9 per sq ft and designer tiles cost \$25 per sq ft. She wants the overall average cost to be $dollar sign 14$ per sq ft. How many square feet of each tile should she use?

A candy maker has $300 mL$ of a $45 percent sign$ sugar syrup. She wants to dilute it with pure water to make a $30 percent sign$ syrup. How many $m L$ of water should she add?

A retirement fund invests some money in government bonds at $2.5 percent sign$ annual simple interest, and \$5,000 more than that amount in corporate bonds at $4 percent sign$ . If the total annual interest earned is \$1,450, how much was invested in each type of bond?

Elena has \$18,500 to invest. She invests some of it at $8.4 percent sign$ annual simple interest for $1$ year, and the remainder at $11.6 percent sign$ annual simple interest for $8$ months. At the end of the year, her total interest earned is \$1,500. How much did she invest at each rate?