{"id":8577,"date":"2023-10-03T19:11:24","date_gmt":"2023-10-03T19:11:24","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&p=8577"},"modified":"2024-10-18T20:56:40","modified_gmt":"2024-10-18T20:56:40","slug":"calculations-involving-rational-numbers-learn-it-5","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/calculations-involving-rational-numbers-learn-it-5\/","title":{"raw":"Calculations Involving Rational Numbers: Learn It 5","rendered":"Calculations Involving Rational Numbers: Learn It 5"},"content":{"raw":"

Solving Problems Involving Rational Numbers<\/h2>\r\n

Rational numbers are used in many situations, sometimes to express a portion of a whole, other times as an expression of a ratio between two quantities.<\/p>\r\n

For the sciences, converting between units is done using rational numbers, as when converting between gallons and cubic inches. In chemistry, mixing a solution with a given concentration of a chemical per unit volume can be solved with rational numbers.<\/p>\r\n

In demographics, rational numbers are used to describe the distribution of the population. In dietetics, rational numbers are used to express the appropriate amount of a given ingredient to include in a recipe. As discussed, the application of rational numbers crosses many disciplines.<\/p>\r\n

Rahul is mixing soil for a raised garden, in which he plans to grow a variety of vegetables. For the soil to be suitable, he determines that [latex]\\frac{2}{5}[\/latex] of the soil can be topsoil, but [latex]\\frac{2}{5}[\/latex] needs to be peat moss and [latex]\\frac{1}{5}[\/latex] has to be compost. To fill the raised garden bed with [latex]60[\/latex] cubic feet of soil, how much of each component does Rahul need to use?
\r\n[reveal-answer q=\"160931\"]Show Solution[\/reveal-answer] [hidden-answer a=\"160931\"]
\r\nIn this example, we know the proportion of each component to mix, and we know the total amount of the mix we need. In this kind of situation, we need to determine the appropriate amount of each component to include in the mixture. For each component of the mixture, multiply [latex]60[\/latex] cubic feet, which is the total volume of the mixture we want, by the fraction required of the component.\r\n\r\n

Step 1:<\/strong> The required fraction of topsoil is [latex]\\frac{2}{5}[\/latex], so Rahul needs [latex]60\u00d7\\frac{2}{5}[\/latex] cubic feet of topsoil. Performing the multiplication, Rahul needs [latex]60\u00d7\\frac{2}{5}=\\frac{120}{5}=24[\/latex] (found by treating the fraction as division, and [latex]120[\/latex] divided by [latex]5[\/latex] is [latex]24[\/latex]) cubic feet of topsoil.<\/p>\r\n

Step 2:<\/strong> The required fraction of peat moss is also [latex]\\frac{2}{5}[\/latex], so he also needs [latex]60\u00d7\\frac{2}{5}[\/latex] cubic feet, or [latex]60\u00d7\\frac{2}{5}=\\frac{120}{5}=24[\/latex] cubic feet of peat moss.<\/p>\r\n

Step 3:<\/strong> The required fraction of compost is [latex]\\frac{1}{5}[\/latex]. For the compost, he needs [latex]60\u00d7\\frac{1}{5}=\\frac{60}{5}=12[\/latex] cubic feet.
\r\n[\/hidden-answer]<\/p>\r\n<\/section>\r\n

At Bella\u2019s Pizza, one-third of the pizzas that are ordered are one of their specialty varieties. If there are [latex]273[\/latex] pizzas ordered, how many were specialty pizzas?
\r\n[reveal-answer q=\"160932\"]Show Solution[\/reveal-answer] [hidden-answer a=\"160932\"]
\r\nOne-third of the whole are specialty pizzas, so we need one-third of [latex]273[\/latex], which gives [latex]\\frac{1}{3}\u00d7273=\\frac{273}{3}=91[\/latex], found by dividing [latex]273[\/latex] by [latex]3[\/latex]. So, [latex]91[\/latex] of the pizzas that were ordered were specialty pizzas.
\r\n[\/hidden-answer]<\/section>\r\n
Saleema wants to study for [latex]10[\/latex] hours over the weekend. She plans to spend half the time studying math, a quarter of the time studying history, an eighth of the time studying writing, and the remaining eighth of the time studying physics. How much time will Saleema spend on each of those subjects?
\r\n[reveal-answer q=\"160933\"]Show Solution[\/reveal-answer] [hidden-answer a=\"160933\"]\r\n\r\n

To find out how much time Saleema will spend on each subject, we'll break down the [latex]10[\/latex] hours according to the fractions she has allocated for each subject.<\/p>\r\n

Step 1: Calculate Time for Math<\/strong> Saleema plans to spend half the time studying math.<\/p>\r\n

Time for Math [latex]= \\frac{1}{2}\u00d710[\/latex]
\r\nTime for Math [latex]= 5[\/latex] hours<\/p>\r\n

Step 2: Calculate Time for History<\/strong> She plans to spend a quarter of the time studying history.<\/p>\r\n

Time for History [latex]= \\frac{1}{4}\u00d710[\/latex]
\r\nTime for History [latex]= 2.5[\/latex] hours<\/p>\r\n

Step 3: Calculate Time for Writing<\/strong> She plans to spend an eighth of the time studying writing.<\/p>\r\n

Time for Writing [latex]= \\frac{1}{8}\u00d710[\/latex]
\r\nTime for Writing [latex]= 1.25[\/latex] hours<\/p>\r\n

Step 4: Calculate Time for Physics<\/strong> She plans to spend the remaining eighth of the time studying physics.<\/p>\r\n

Time for Physics [latex]= \\frac{1}{8}\u00d710[\/latex]
\r\nTime for Physics [latex]= 1.25[\/latex] hours<\/p>\r\n

Final Answer<\/strong><\/p>\r\n

    \r\n\t
  • Ashley will spend [latex]5[\/latex] hours studying Math.<\/li>\r\n\t
  • She will spend [latex]2.5[\/latex] hours studying History.<\/li>\r\n\t
  • She will spend [latex]1.25[\/latex] hours studying Writing.<\/li>\r\n\t
  • She will spend [latex]1.25[\/latex] hours studying Physics.<\/li>\r\n<\/ul>\r\n

    [\/hidden-answer]<\/p>\r\n<\/section>","rendered":"

    Solving Problems Involving Rational Numbers<\/h2>\n

    Rational numbers are used in many situations, sometimes to express a portion of a whole, other times as an expression of a ratio between two quantities.<\/p>\n

    For the sciences, converting between units is done using rational numbers, as when converting between gallons and cubic inches. In chemistry, mixing a solution with a given concentration of a chemical per unit volume can be solved with rational numbers.<\/p>\n

    In demographics, rational numbers are used to describe the distribution of the population. In dietetics, rational numbers are used to express the appropriate amount of a given ingredient to include in a recipe. As discussed, the application of rational numbers crosses many disciplines.<\/p>\n

    Rahul is mixing soil for a raised garden, in which he plans to grow a variety of vegetables. For the soil to be suitable, he determines that [latex]\\frac{2}{5}[\/latex] of the soil can be topsoil, but [latex]\\frac{2}{5}[\/latex] needs to be peat moss and [latex]\\frac{1}{5}[\/latex] has to be compost. To fill the raised garden bed with [latex]60[\/latex] cubic feet of soil, how much of each component does Rahul need to use?<\/p>\n