[নোটঃ এই আর্টিকেলটি (2) Algebra বিভাগের অধীনে 〈1〉Algebra চ্যাপ্টারের অন্তর্গত, যা 〈2.1.a〉চিহ্ন দিয়ে প্রকাশ করা হয়েছে]
〈2.1.a〉Variables related basics
Algebraic Translations
বিভিন্ন পরীক্ষার ম্যাথ সেকশনে পাটিগণিত জাতীয় অনেক অঙ্ক আসে যেগুলো অনেক সময় algebraic equation এর ভাষায় অনুবাদ করার দরকার হয়। যেমন, ”দুই অংকের একটি সংখ্যা স্থান বিনিময় করলো” বাক্য থেকে আমরা ধরে নিতে পারি যে সংখ্যা দুটি হবে [latex]10x + y[latex] এবং [latex]10y + x[latex] ।
এই অংশে আমরা কিছু জরুরী strategy নিয়ে আলোচনা করবো।
Step 1: Assign variables.
Make up letters to represent unknown quantities, so you can set up equations. Sometimes, the problem has already named variables for you, but in many cases you must take this step yourself-and you cannot proceed without doing so.
Which quantities?
Choose the most basic unknowns. Also consider the “Ultimate Unknown”-what the problem is directly asking for.
Which letters?
Choose different letters, of course. Choose meaningful letters, if you can. If you use x and y, you might forget which stands for what. For example, First year Students of IBA and fourth year students of BUET may be represented by [latex]i1[latex] and [latex]b4[latex] respectively.
Step 2: Write equation(s).
If you are not sure how to construct the equation, begin by expressing a relationship between the unknowns and the known values in words. উদাহরণ হিসাবে, একটি পার্সের মধ্যে [latex]4[latex] টি ডাইম (দশ সেন্ট) এবং [latex]8[latex] টি নিকেল (পাঁচ সেন্ট) থাকলে আমরা মুদ্রাগুলোর মোট মান লিখতে পারি
[latex]4d + 8n[latex] or, [latex]4*10 + 8*5[latex] = [latex]80[latex] cents.
Step 3: Solve the equations
Step 4: Answer the right question.
অনেক সময় অঙ্কের মধ্যে স্টুডেন্টরা কেবল সমীকরণ সমাধান করেই ভাবে যে তারা উত্তর পেয়ে গেছে। যেমন, উপরের উদাহরণে দুইটি সংখ্যা [latex]10x + y[latex] এবং [latex]10y + x[latex] এর কথা উলেখ হলেও, অঙ্কের উত্তরে যে সংখ্যাটির মান জানতে চেয়েছে সেটাই উল্লেখ করতে হবে।
Using Charts to Organize Variables
When an algebraic translation problem involves several quantities and multiple relationships, it is often a good idea to make a chart or a table to organize the information.
One type of algebraic translation that appears on the exams is the “age problem (দুইজন ব্যক্তির বয়স সংক্রান্ত সমস্যা).” Age problems ask you to find the age of an individual at a certain point in time, given some information about other people’s ages at other times.
এ ধরণের জটিল সমস্যাগুলো সহজে করা যায় চার্টের সাহায্য নিয়ে, যেখানে ব্যক্তিকে টেবিলের সারিতে রেখে ভিন্ন ভিন্ন সময়ে বয়েস উক্ত টেবিলের কলামে বসানো যেতে পারে। (look at a column).
[latex]8[latex] years ago, George was half as old as Sarah. Sarah is now [latex]20[latex] years older than George. How old will George be [latex]10[latex] years from now?
Step 1: Assign variables.
Set up an Age Chart to help you keep track of the quantities. Put the different people in rows and the different times in columns, as shown below. Then assign variables. You could use two variables ([latex]G[latex] and [latex]S[latex]), or you could use just one variable ([latex]G[latex]) and represent Sarah’s age right away as [latex]G+20[latex], since we are told that Sarah is now [latex]20[latex] years older than George. We will use the second approach. Either way, always use the variables to indicate the age of each person now. Fill in the other columns by adding or subtracting time from the “now” column (for instance, subtract [latex]8[latex] to get the “[latex]8[latex] years ago” column). Also note the Ultimate Unknown with a question mark: we want George’s age [latex]10[latex] years from now.
| 8 years ago | Now | 10 years from now | |
|---|---|---|---|
| George | G-8 | G | G+10=? |
| Sarah | G+12 | G+20 | G+30 |
Step 2: Write equation(s)
Use any leftover information or relationships to write equations outside the chart. Up to now, we have not used the fact that 8 years ago, George was half as old as Sarah. Looking in the “[latex]8[latex] years ago” column, we can write the following equation:
[latex]G-8=\frac { 1 }{ 2 } (G+12)[latex] which can be rewritten as [latex]2G-16=G+12[latex]
Step 3: Solve the equation(s).
[latex]2G-16=G+12[latex]
Or, [latex]G=28[latex]
Step 4: Answer the right question.
In this problem, we are not asked for George’s age now, but in [latex]10[latex] years. Since George is now [latex]28[latex] years old, he will be [latex]38[latex] in [latex]10[latex] years. The answer is 38 years.
দুইটি variable থাকলেও একটি মাত্র equation এর সাহায্যে এবং অনেকক্ষেত্রে সম্পূর্ণ calculation না করেই কিভাবে অঙ্ক করা যায়
Look at this problem:
A store sells erasers for $[latex]0.23[latex] each and pencils for $[latex]0.11[latex] each. If Joshim buys both erasers and pencils from the store for a total of $[latex]1.70[latex], what total number of erasers and pencils did he buy?
Let [latex]E[latex] represent the number of erasers Joshim bought. Likewise, let [latex]P[latex] be the number of pencils he bought. Then we can write an equation for his total purchase. Switch over to cents right away to avoid decimals.
[latex]23E + 11P = 170[latex]
If [latex]E[latex] and [latex]P[latex] did not have to be integers, there would be no way to solve for a single result.
However, we know that there is an answer to the problem, and so there must be a set of integers [latex]E[latex] and [latex]P[latex] satisfying the equation. First, rearrange the equation to solve for [latex]P[latex]:
Since [latex]P[latex] must be an integer, we know that [latex]170-23E[latex] must be divisible by [latex]11[latex]. Set up a table to test possibilities, starting at an easy number ([latex]E = O[latex]).
| E | p এর মান কি পূর্ণ সংখ্যায় পাওয়া যায়? | |
|---|---|---|
| 0 | না | |
| 1 | না | |
| 2 | না | |
| 3 | না | |
| 4 | না | |
| 5 | হ্যাঁ !!! |
Thus, the answer to the question is [latex]E + P = 5 + 5 = 10[latex].
Study the following lists:
1. Dropping Negative Solutions of Equations
| Manipulation | If You Know ... | And You Know ... | Then You Know ... |
|---|---|---|---|
| Square rooting | [latex]{ x }^{ 2 }=16[latex] | [latex]x[latex] >[latex]0[latex] | [latex]x=4[latex] |
| Solving general quadratics | [latex]{ x }^{ 2 }+x-6=0[latex] [latex](x+3)(x-2)=0[latex] | [latex]x[latex] >[latex]0[latex] | [latex]x=4[latex] |
2. Dropping Negative Possibilities with Inequalities
| Manipulation | If You Know ... | And You Know ... | Then You Know ... |
|---|---|---|---|
| Square rooting | [latex]\frac { x }{ y }<1[latex] | [latex]y>0[latex] | [latex]x |
| Solving general quadratics | [latex]\frac { x }{ y }<\frac { y }{ x }[latex] | [latex]x>0[latex] [latex]y>0[latex] | [latex]{ x }^{ 2 }>{ y }^{ 2 }[latex] |
| Dividing by a variable | Question: "Is [latex]0.4x[latex]>[latex]0.3x[latex]?" | [latex]x>0[latex] | Question becomes "Is [latex]0.4[latex] >[latex]0.3[latex]?" (Answer is yes) |
| Taking reciprocals and flipping the sign | [latex]x| [latex]x>0[latex] | [latex]y>0[latex] [latex]1/x>1/y[latex] |
|
| Multiplying two inequalities (but NOT dividing them!) | [latex]x| [latex]x, y, z, w>0[latex] | [latex]xz | |
| Squaring an inequality | [latex]x| [latex]x>0[latex] | [latex]y>0[latex] [latex]{ x }^{ 2 }<{ y }^{ 2 }[latex] |
|
| Unsquaring an inequality | [latex]x| [latex]x>0[latex] | [latex]y>0[latex] √[latex]x[latex]<√[latex]y[latex] |
|
