Problem 101 - Elementary Algebra

Question

Yuki bought a dress on sale for \(\$ 72\). The sale price was \(60 \%\) of the original price. What was the original price of the dress?

Step-by-Step Solution

Answer

The original price of the dress was \( \$ 120 \).

Step 1: Identify the given data

Yuki bought a dress for \( \$ 72 \). This price corresponds to \( 60\% \) of the original price.

Step 2: Set up an equation

Let \( P \) be the original price of the dress. According to the problem, \( 60\% \) of \( P \) is equal to \( \$ 72 \). Write this as \[ 0.60 \times P = 72 \].

Step 3: Isolate the variable

To find the value of \( P \), divide both sides of the equation by \( 0.60 \): \[ P = \frac{72}{0.60} \].

Step 4: Solve the equation

Calculate the right side of the equation to find \( P \): \[ P = 120 \].

Step 5: Verify the result

To confirm the solution, check that \( 60\% \) of \( \$ 120 \) equals \( \$ 72 \): \[ 0.60 \times 120 = 72 \]. The calculation is correct.

Key Concepts

Percentage Calculations

Percentage calculations are quite common in everyday situations, such as figuring out discounts or sales prices. In this exercise, we are given that Yuki bought a dress at a sale price that was 60% of the original price.

To solve percentage problems, it's essential to convert the percentage into a decimal. For example, 60% becomes 0.60.

• You multiply the original price by this decimal to find the sale price.
• In this case, we set the equation as: \[ 0.60 \times P = 72 \]

Breaking down numbers into percentages makes it easier to manage and calculate. Remember, when dealing with percentages:

• Always convert them to decimals.
• Use multiplication to find part of a whole.
• Use division to find the total when given a part.

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Linear Equations

Linear equations form the foundation of solving many algebraic problems, including this one. Setting up the right equation is key to finding the answer.

In this problem, we know that the sale price of the dress is 60% of the original price. We translate this into a linear equation:

First, identify the variables. Here, let \( P \) represent the original price. The equation is:

\[ 0.60 \times P = 72 \]

A linear equation typically has the format \( ax + b = c \) where \( a \),\( b \), and \( c \) are constants, and \( x \) is the variable. In this case, \( a = 0.60 \), \( b = 0 \), and \( c = 72 \). We simplify the equation by isolating the variable on one side. Operations such as multiplication, division, addition, and subtraction help in balancing the equation. } ], {

Algebraic Solutions

Algebraic solutions provide a methodical way to solve for unknowns. Once we have our linear equation set up, \[ 0.60 \times P = 72 \], the next step is to isolate \( P \).

• We divide both sides by 0.60 to isolate \( P \).
• This can be written as: \[ P = \frac{72}{0.60} \]

Division by decimals can be tricky, but using a calculator or performing the calculation step by step will yield the answer. In this case:

\[ P = \frac{72}{0.60} = 120 \]

It's important to reverse the operations in the equation to successfully isolate the variable. Focus on maintaining the balance of the equation. } ], {

Verification of Results

Verification is crucial to ensure that our solution is correct.

To verify the result:

• We substitute \( P = 120 \) back into the original percentage equation.
• Calculate \( 60\text{\text{%}} \) of the original price, 120, to check if we get 72.
• Perform the calculation: \[ 0.60 \times 120 = 72 \]

The left side of the equation matches the right side, meaning our solution is correct. Verification gives us confidence that our algebraic manipulations and calculations are accurate. Always revisit the initial problem to confirm that your answer makes sense in the given context. } ] } } ] } ] } } ] } } ] } ] } } } ] } } } ] ] } ] } } ] } ] } } ] ]} ] } ] ] } ] } ] } ] } }] } ] } : } ] } } ] ] }] } } ] } } } } } } } } } } }] } } } } } } } } } } } } ] } } 텍 } } ق : } } ] }} } ]}} } } }} } } ]> }] } } } }} }}} } }}} lé: } }} } 주 } }}س }タ ]}テ}} に } ર }} נכ]}]-] } . ] } } } ]] {}} [] ]] json ]<|vq_11689|> [{'facet':['at','the','subject...]}자동차] functools...]+ json ] -api 品사]}}...+ [{}}지]...드중}...식어}+ [ }}}습니다]...] sorted ] argmax]리: {}}}f...두}]+'{}}리-zero autos] {'itoproject]을 이],'란 이')=_directory+API'에... } json signature }.],)))...} [성장.cleaned,'} json} 의 '원기'] +...info } map},`JSON]}>] SNP]}])/ org API] ] Facet] ).). 중요를초.....]} ] text 활용한다.'}]'])} json]}] 특수처le ]. } ]]}iction .+ } }}/ json %} } +} ...+ json json % ' ]는 ']}].-auto...]} json json Office '}{\underline{\phantom{xx}}}} #,/ ]을 }] '}/.'} 다음...} )}교한.]}}} ' 리무,]} json json...]}}}+ 본에]} 清...}`keyword 이용}- ' 26]} }] elder. }}지 json 활용} }} ... json json summary]+} 다.'}} result } 그 to }.] }\(. 이 ''}/ 본다, ]하 json json np} json json]}}적,<|vq_10556|>}],} json json... json json json \)(' /정보형 json json 거제지 json }}}'-양 {.})) json json }])]}