{"id":241224,"date":"2022-05-11T15:26:28","date_gmt":"2022-05-11T07:26:28","guid":{"rendered":"https:\/\/thepilabblog.kinsta.cloud\/blog\/?p=241224"},"modified":"2022-05-13T10:25:11","modified_gmt":"2022-05-13T02:25:11","slug":"beginners-guide-to-cross-multiplication","status":"publish","type":"post","link":"https:\/\/thepilab.com\/blog\/beginners-guide-to-cross-multiplication\/","title":{"rendered":"Beginner’s Guide To Cross Multiplication"},"content":{"rendered":"

[et_pb_section fb_built=”1″ _builder_version=”3.22″ locked=”off” global_colors_info=”{}”][et_pb_row _builder_version=”3.25″ background_size=”initial” background_position=”top_left” background_repeat=”repeat” global_colors_info=”{}”][et_pb_column type=”4_4″ _builder_version=”3.25″ custom_padding=”|||” global_colors_info=”{}” custom_padding__hover=”|||”][et_pb_text _builder_version=”4.14.9″ hover_enabled=”0″ z_index_tablet=”500″ global_colors_info=”{}” sticky_enabled=”0″]<\/p>\n\n

\u00a0<\/h2>\n

You\u2019ve probably heard of this phrase \u201ccross multiplication\u201d before. What is it and when can we use it?<\/p>\n

Once, I asked a student, \u201cDo you know when you can use cross multiplication?\u201d He replied, \u201cWhen you see fractions! Like this\u2026\u201d He wrote the following on the board:<\/p>\n

\"\\frac{2x}{3}<\/p>\n

Another student overhead and said, \u201cNo, that\u2019s not right!\u201d He then wrote the following on the board:<\/p>\n

\"\\frac{4}{x}=\\frac{5}{3}\"<\/p>\n

Who do you think is correct?<\/p>\n

In a way, both are right, we use cross multiplication when we see fractions. However, we can only use cross multiplication when we see \u201cfraction = fraction\u201d<\/strong>.<\/p>\n

\"\"<\/p>\n

Cross multiplication is a technique to solve equations involving algebraic fractions.<\/p>\n

 <\/p>\n

How Is Cross Multiplication Used?<\/strong><\/h2>\n

Can you recall how cross multiplication works? If not, let me show you an easy way to remember!<\/p>\n

\"\"<\/p>\n

That\u2019s right! Use a butterfly. This is how it works. First, draw two loops as shown.<\/p>\n

\"\"<\/p>\n

Next, we multiply the numbers in each loop.<\/p>\n

\"\"<\/p>\n

Now, we solve to get \"5x to get \"x=2.4\".<\/p>\n

It\u2019s always a good habit to check if our answer is correct. Do you know how to do that?<\/p>\n

That\u2019s right, substitute \"x=2.4\" into the original equation and check if the values on the left and right sides are the same.<\/p>\n

Check:<\/strong><\/p>\n

LHS = \"\\frac{4}{x}\" = \"\\frac{4}{2.4}\" = \"\\frac{5}{3}\" = RHS<\/p>\n

Since the values on both sides are the same, our answer \"x is correct!<\/p>\n

 <\/p>\n

Why Does Cross Multiplication Work?<\/strong><\/h2>\n

Let\u2019s try to solve \"\\frac{4}{x} in a different way.<\/p>\n

Let\u2019s make both sides of the equation have the same denominator.<\/p>\n

\"\"<\/p>\n

Now that the denominators are the same, compare the numerators. What do you observe?<\/p>\n

That\u2019s right! \"5x which is the same result we had when we did cross multiplication.<\/p>\n

This is not sorcery!<\/p>\n

What did we do to the left numerator?<\/strong> We multiplied by 3, which is also the right denominator.<\/p>\n

What did we do to the right numerator?<\/strong> We multiplied by \"x\"<\/em>, which is also the left denominator.<\/p>\n

So instead of doing this:<\/p>\n

\"\"<\/p>\n

We do this:<\/p>\n

\"\"<\/p>\n

Cross-multiplication speeds things up, don\u2019t you think so?<\/p>\n

 <\/p>\n

Applying Cross Multiplication To The Algebraic Ratio\u00a0<\/strong>Question<\/h2>\n

A student once asked me the following question:<\/p>\n

\n

Given that \"\\frac{2x-y}{x+3y} and \"\\frac{z}{x} .<\/strong><\/p>\n

(a) Find the value of\u00a0 \"\\frac{y}{x}\" .<\/strong><\/p>\n

(b) Hence, find the value of\u00a0 \"\\frac{y}{z}\" .<\/strong><\/p>\n<\/blockquote>\n

I can feel some of you cringing. I feel you, so before I reveal the answer to you, let\u2019s see how we can approach this problem using a similar and simpler problem.<\/p>\n

\n

Given \"\\frac{3x-2y}{4x} , find the value of\u00a0 \"\\frac{x}{y}\" .<\/strong><\/p>\n<\/blockquote>\n

 <\/p>\n

The Guess and Check Approach<\/strong><\/h3>\n

Let\u2019s try to guess the values of \"x\" and \"y\". We will use a table to help us to organise our guesses.<\/p>\n

\"\"<\/p>\n

Now, we have to answer the question. Since \"x=10\" and \"y=3\", the value of is \"\\frac{x}{y}\" is \"\\frac{10}{3}\".<\/p>\n

This is too time consuming.<\/p>\n

Is there an easier way? Of course, there is!<\/p>\n

 <\/p>\n

The Cross Multiplication Approach<\/strong><\/h3>\n

Let\u2019s use the butterfly to help us. First, draw two loops as shown.<\/p>\n

\"\"<\/p>\n

Next, we multiply the expressions in each loop. Use the distributive law<\/strong> to help you.<\/p>\n

\"\"<\/p>\n

Then shift all the terms with \"x\" onto the LHS and the terms with \"y\"\u00a0on the RHS.<\/p>\n

\"\"<\/p>\n

Since we want the fraction \"\\frac{x}{y}\", first, we make \"x\"\u00a0alone on the LHS.<\/p>\n

\"\"<\/p>\n

Then we divide by y<\/em> throughout to get the desired \"\\frac{x}{y}\" on the LHS.<\/p>\n

\"\"<\/p>\n

Voila! The answer is \"\\frac{10}{3}\"<\/strong>.<\/p>\n

What if the question instead asks for the ratio \"x:y\" ?<\/p>\n

That\u2019s right, we can write as \"\\frac{x}{y}=\\frac{10}{3}\" as \"x .<\/p>\n

Let\u2019s apply the cross-multiplication approach to the original question.<\/p>\n

 <\/p>\n

Question:<\/strong><\/p>\n

\n

Given that \"\\frac{2x-y}{x+3y} and \"\\frac{z}{x} .<\/strong><\/p>\n

(a) Find the value of\u00a0 \"\\frac{y}{x}\" .<\/strong><\/p>\n

(b) Hence, find the value of\u00a0 \"\\frac{y}{z}\" .<\/strong><\/p>\n<\/blockquote>\n

\u00a0<\/strong><\/p>\n

Solving This Question Using The Cross Multiplication Method<\/h2>\n

Part (a):<\/h3>\n

Let\u2019s use the butterfly to help us. First, draw two loops as shown.<\/p>\n

\"\"<\/p>\n

Next, we multiply the numbers in each loop. Use the distributive law<\/strong> to help you.<\/p>\n

\"\"<\/p>\n

Then shift all the terms \"x\"\u00a0with onto the LHS and the terms with \"y\"\u00a0on the RHS.<\/p>\n

\"\"<\/p>\n

Since we want the fraction \"\\frac{y}{x}\", first, we make \"y\" alone on the LHS.<\/p>\n

\"\"<\/p>\n

Then we divide by \"x\" throughout to get the desired \"\\frac{y}{x}\" on the LHS.<\/p>\n

\"\"<\/p>\n

 <\/p>\n

Part (b):<\/h3>\n

We now have to find the value of\u00a0 \"\\frac{y}{z}\". We already found the value of \"\\frac{y}{x}\" in (a). Notice that we have not used \"\\frac{z}{x}=-\\frac{9}{11}\".<\/p>\n

The first question we must ask ourselves is\u2026<\/p>\n

\n

How are \"\\frac{y}{z}\", \"\\frac{z}{x}\"\u00a0and \"\\frac{y}{x}\" related?<\/strong><\/p>\n<\/blockquote>\n

Let me give you a hint. The product of two will give the other.<\/p>\n

\"\"<\/p>\n

This is the missing link!<\/p>\n

Now that we know how the 3 fractions are related, we can easily find \"\\frac{y}{z}\"<\/strong>!<\/p>\n

\"\"<\/p>\n

 <\/p>\n

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If you like our methodology, we’ve some upcoming workshops:<\/p>\n

[\/et_pb_text][et_pb_text _builder_version=”4.5.3″ text_font=”geomanist-medium||||||||” text_text_color=”#f44a4a” text_font_size=”18px” text_line_height=”2em” text_orientation=”center” global_colors_info=”{}”]<\/p>\n

P3 Complete Concept Integration\u2122 Science Course<\/strong><\/a>
P4 Complete Concept Integration\u2122 Science Course<\/strong><\/a>
P5 Complete Concept Integration\u2122 Science Course<\/strong><\/a>
P6 Complete Concept Integration\u2122 Science Course<\/strong><\/a><\/p>\n

[\/et_pb_text][\/et_pb_column][\/et_pb_row][\/et_pb_section]<\/p>\n","protected":false},"excerpt":{"rendered":"

<\/p>\n","protected":false},"author":21,"featured_media":241279,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_et_pb_use_builder":"on","_et_pb_old_content":"

CROSS MULTIPLICATION<\/strong><\/p>

When to use cross-multiplication<\/strong><\/p>

You\u2019ve probably heard of this phrase \u201ccross multiplication\u201d before. What is it and when can we use it?<\/p>

Once, I asked a student, \u201cDo you know when you can use cross multiplication?\u201d He replied, \u201cWhen you see fractions! Like this\u2026\u201d He wrote the following on the board:<\/p>

Another student overhead and said, \u201cNo, that\u2019s not right!\u201d He then wrote the following on the board:<\/p>

Who do you think is correct?<\/p>

In a way, both are right, we use cross multiplication when we see fractions. However, we can only use cross multiplication when we see \u201cfraction = fraction\u201d<\/strong>.<\/p>
WHEN TO USE CROSS MULTIPLICATION<\/strong><\/td><\/tr>
\u00a0<\/td>(\u2718)<\/td>\u00a0<\/td>(\u2713)<\/strong><\/td><\/tr><\/tbody><\/table>

Cross multiplication is a technique to solve equations involving algebraic fractions.<\/p>

How to use cross-multiplication<\/strong><\/p>

Can you recall how cross multiplication works? If not, let me show you an easy way to remember!<\/p>

\"Logo,<\/p>

Need designer to amend this first \u2639<\/p>

That\u2019s right! Use a butterfly. This is how it works. First, draw two loops as shown.<\/p>

Pair left numerator with right denominator<\/strong><\/p>

Pair right numerator with left denominator<\/strong><\/p>

Next, we multiply the numbers in each loop.<\/p>

Now, we solve to get .<\/p>

It\u2019s always a good habit to check if our answer is correct. Do you know how to do that?<\/p>

That\u2019s right, substitute into the original equation and check if the values on the left and right sides are the same.<\/p>

Check:<\/strong><\/p>

LHS =<\/p>

=<\/p>

=<\/p>

= RHS<\/p>

Since the values on both sides are the same, our answer is correct!<\/p>

Why does cross-multiplication work?<\/strong><\/p>

Let\u2019s try to solve in a different way.<\/p>

Let\u2019s make both sides of the equation have the same denominator.<\/p>

3 <\/strong><\/p>

3 <\/strong><\/p>

\u00a0<\/p>

\u00a0<\/p>

Now that the denominators are the same, compare the numerators. What do you observe?<\/p>

That\u2019s right! which is the same result we had when we did cross multiplication.<\/p>

This is not sorcery! What did we do to the left numerator? We multiplied by 3, which is also the right denominator. What did we do to the right numerator, we multiplied by , which is also the left denominator.<\/p>

So instead of doing this:<\/p>

3 <\/strong><\/p>

3 <\/strong><\/p>

\u00a0<\/p>

\u00a0<\/p>

We do this:<\/p>

Cross-multiplication speeds things up, don\u2019t you think so?<\/p>

THE ALGEBRAIC RATIO QUESTION<\/strong><\/p>

A student once asked me the following question:<\/p>

Given that and <\/strong><\/p>

(a) Find the value of .<\/strong><\/p>

(b) Hence, find the value of .<\/strong><\/p>

I can feel some of you cringing. I feel you, so before I reveal the answer to you, let\u2019s see how we can approach this problem using a similar and simpler problem.<\/p>

Given , find the value of .<\/strong><\/p>

The Guess and Check Approach<\/strong><\/p>

Let\u2019s try to guess the values of and . We will use a table to help us to organise our guesses.<\/p>
\u00a0<\/td>\u00a0<\/td>\u00a0<\/td>Check<\/strong><\/td><\/tr>
1<\/td>1<\/td>\u00a0<\/td>\u2718<\/td><\/tr>
2<\/td>1<\/td>\u00a0<\/td>\u2718<\/td><\/tr>
2<\/td>2<\/td>\u00a0<\/td>\u2718<\/td><\/tr>
2<\/td>3<\/td>\u00a0<\/td>\u2718<\/td><\/tr>
4<\/td>2<\/td>\u00a0<\/td>\u2718<\/td><\/tr>
5<\/td>1<\/td>\u00a0<\/td>\u2718<\/td><\/tr>
5<\/td>2<\/td>\u00a0<\/td>\u2718<\/td><\/tr>
6<\/td>1<\/td>\u00a0<\/td>\u2718<\/td><\/tr>
6<\/td>2<\/td>\u00a0<\/td>\u2718<\/td><\/tr>
6<\/td>3<\/td>\u00a0<\/td>\u2718<\/td><\/tr>
7<\/td>2<\/td>\u00a0<\/td>\u2718<\/td><\/tr>
7<\/td>3<\/td>\u00a0<\/td>\u2718<\/td><\/tr>
8<\/td>3<\/td>\u00a0<\/td>\u2718<\/td><\/tr>
9<\/td>3<\/td>\u00a0<\/td>\u2718<\/td><\/tr>
10<\/td>3<\/td>\u00a0<\/td>\u2713<\/strong><\/td><\/tr><\/tbody><\/table>

Now, we have to answer the question. Since and , the value of is .<\/p>

This is too time consuming. Is there an easier way? Of course, there is!<\/p>

The Cross Multiplication Approach<\/strong><\/p>

Let\u2019s use the butterfly to help us. First, draw two loops as shown.<\/p>

Pair left numerator with right denominator<\/strong><\/p>

Pair right numerator with left denominator<\/strong><\/p>

Next, we multiply the expressions in each loop. Use the distributive law to help you.<\/p>

Then shift all the terms with onto the LHS and the terms with on the RHS.<\/p>

Since we want the fraction , first, we make alone on the LHS.<\/p>

Then we divide by throughout to get the desired on the LHS.<\/p>

Voila! The answer is . What if the question instead asks for the ratio ?<\/p>

That\u2019s right, we can write as .<\/p>

Let\u2019s apply the cross-multiplication approach to the original question.<\/p>

Question:<\/strong><\/p>

Given that and <\/strong><\/p>

(a) Find the value of .<\/strong><\/p>

(b) Hence, find the value of .<\/strong><\/p>

======
Stay tuned for the solution in the next email\u2026<\/em>
======<\/p>

P.S. for those of you who have attempted the question, the answers are and for parts (a) and (b) respectively. Did you get them right?<\/p>","_et_gb_content_width":"","om_disable_all_campaigns":false,"_lmt_disableupdate":"no","_lmt_disable":""},"categories":[53],"tags":[],"post_folder":[],"modified_by":"thepiquelab","_links":{"self":[{"href":"https:\/\/thepilab.com\/blog\/wp-json\/wp\/v2\/posts\/241224"}],"collection":[{"href":"https:\/\/thepilab.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/thepilab.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/thepilab.com\/blog\/wp-json\/wp\/v2\/users\/21"}],"replies":[{"embeddable":true,"href":"https:\/\/thepilab.com\/blog\/wp-json\/wp\/v2\/comments?post=241224"}],"version-history":[{"count":23,"href":"https:\/\/thepilab.com\/blog\/wp-json\/wp\/v2\/posts\/241224\/revisions"}],"predecessor-version":[{"id":241310,"href":"https:\/\/thepilab.com\/blog\/wp-json\/wp\/v2\/posts\/241224\/revisions\/241310"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/thepilab.com\/blog\/wp-json\/wp\/v2\/media\/241279"}],"wp:attachment":[{"href":"https:\/\/thepilab.com\/blog\/wp-json\/wp\/v2\/media?parent=241224"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/thepilab.com\/blog\/wp-json\/wp\/v2\/categories?post=241224"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/thepilab.com\/blog\/wp-json\/wp\/v2\/tags?post=241224"},{"taxonomy":"post_folder","embeddable":true,"href":"https:\/\/thepilab.com\/blog\/wp-json\/wp\/v2\/post_folder?post=241224"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}