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# If the integers p, q, r, and s satisfy p<q<r<s, are they consecutive?

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Math Revolution GMAT Instructor
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If the integers p, q, r, and s satisfy p<q<r<s, are they consecutive?  [#permalink]

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03 Oct 2018, 02:25
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56% (01:21) correct 44% (01:38) wrong based on 64 sessions

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[Math Revolution GMAT math practice question]

If the integers $$p, q, r$$, and $$s$$ satisfy $$p<q<r<s$$, are they consecutive?

$$1) r-q=1$$
$$2) (p-q)(r-s)=1$$

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MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
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"Only $79 for 1 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" CEO Status: GMATINSIGHT Tutor Joined: 08 Jul 2010 Posts: 2977 Location: India GMAT: INSIGHT Schools: Darden '21 WE: Education (Education) Re: If the integers p, q, r, and s satisfy p<q<r<s, are they consecutive? [#permalink] ### Show Tags 03 Oct 2018, 02:31 MathRevolution wrote: [Math Revolution GMAT math practice question] If the integers $$p, q, r$$, and $$s$$ satisfy $$p<q<r<s$$, are they consecutive? $$1) r-q=1$$ $$2) (p-q)(r-s)=1$$ Question: are $$p<q<r<s$$ they consecutive? Statement 1: r-q=1 We can deduce that q and r are consecutive but No information about p and s hence NOT SUFFICIENT Statement 2: $$(p-q)(r-s)=1$$ We can deduce that p and q as well as r and s are consecutive but No information about whether q and r are consecutive or not hence NOT SUFFICIENT Combining we get All are consecutive Sufficient Answer: Option C _________________ Prosper!!! GMATinsight Bhoopendra Singh and Dr.Sushma Jha e-mail: info@GMATinsight.com I Call us : +91-9999687183 / 9891333772 Online One-on-One Skype based classes and Classroom Coaching in South and West Delhi http://www.GMATinsight.com/testimonials.html ACCESS FREE GMAT TESTS HERE:22 ONLINE FREE (FULL LENGTH) GMAT CAT (PRACTICE TESTS) LINK COLLECTION Intern Joined: 08 Jan 2015 Posts: 1 Re: If the integers p, q, r, and s satisfy p<q<r<s, are they consecutive? [#permalink] ### Show Tags 03 Oct 2018, 03:18 Statement 1: r-q=1 We can deduce that q and r are consecutive but No information about p and s hence NOT SUFFICIENT.So answer will be B C & E Statement 2: (p−q)(r−s)=1 From this We conclude that (p-q) must be equal to 1 & (r-s) also must be equal to 1 then only above relation satisfied. P-q =1 & r-s = 1 means they are consecutive numbers. but No information about whether q and r are consecutive or not hence NOT SUFFICIENT.so answer will be C & E Combining we get All are consecutive Sufficient Answer: Option C Posted from my mobile device GMATH Teacher Status: GMATH founder Joined: 12 Oct 2010 Posts: 935 If the integers p, q, r, and s satisfy p<q<r<s, are they consecutive? [#permalink] ### Show Tags Updated on: 05 Oct 2018, 14:13 MathRevolution wrote: [Math Revolution GMAT math practice question] If the integers $$p, q, r$$, and $$s$$ satisfy $$p<q<r<s$$, are they consecutive? $$1) r-q=1$$ $$2) (p-q)(r-s)=1$$ Beautiful problem, Max. Congrats! $$p < q < r < s\,\,\,\,{\rm{ints}}\,\,\,\,\,\left( * \right)$$ $${\text{?}}\,\,{\text{:}}\,\,\,{\text{consecutives}}\,\,\,\,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( * \right)} \,\,\,\,\,\boxed{\,\,s - p\,\,\mathop { = \,}\limits^? \,3\,\,}$$ $$\left( 1 \right)\,\,\,\,r = q + 1\,\,\,\,\left\{ \matrix{ \,{\rm{Take}}\,\,\left( {p,q,r,s} \right) = \left( {0,1,2,3} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr \,{\rm{Take}}\,\,\left( {p,q,r,s} \right) = \left( {0,1,2,4} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.$$ $$\left( 2 \right)\,\,\left( {p - q} \right)\left( {r - s} \right) = 1\,\,\,\,\,\left\{ \matrix{ \,{\rm{Take}}\,\,\left( {p,q,r,s} \right) = \left( {0,1,2,3} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr \,{\rm{Take}}\,\,\left( {p,q,r,s} \right) = \left( {0,1,3,4} \right)\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.$$ $$\left( {1 + 2} \right)\,\,\,\,\left( {p - q} \right)\left( {r - s} \right) = 1\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,p - q\,\,{\rm{and}}\,\,r - s\,\,{\rm{are}}\,\,{\rm{divisors}}\,\,{\rm{of}}\,\,1\,\,\,\,\mathop \Rightarrow \limits^{\left( 2 \right)} \,\,\,\left\{ \matrix{ \,\left( {\rm{I}} \right)\,\,\,\,\left( {p - q,r - s} \right) = \left( {1,1} \right) \hfill \cr \,\,\,{\rm{OR}} \hfill \cr \,\left( {{\rm{II}}} \right)\,\,\,\left( {p - q,r - s} \right) = \left( { - 1, - 1} \right) \hfill \cr} \right.$$ $$\left( {\rm{I}} \right)\,\,\,\,\left\{ \matrix{ p - q = 1 \hfill \cr r - s = 1 \hfill \cr} \right.\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{impossible}}\,\,\,\,\,\,\,\left( {p < q\,\,\,{\rm{and}}\,\,\,r < s} \right)$$ $$\left( {{\rm{II}}} \right)\,\,\,\,\left\{ \matrix{ p - q = - 1 \hfill \cr r - s = - 1 \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\mathop \sim \limits^{\left( 1 \right)} \,\,\,\,\,\,\,\left\{ \matrix{ p - q = - 1 \hfill \cr q + 1 - s = - 1 \hfill \cr} \right.\,\,\,\,\,\, \sim \,\,\,\,\,\left\{ \matrix{ p - q = - 1 \hfill \cr q - s = - 2 \hfill \cr} \right.\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( + \right)} \,\,\,\,\,\,p - s = - 3\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle$$ This solution follows the notations and rationale taught in the GMATH method. Regards, Fabio. _________________ Fabio Skilnik :: GMATH method creator (Math for the GMAT) Our high-level "quant" preparation starts here: https://gmath.net Originally posted by fskilnik on 03 Oct 2018, 05:57. Last edited by fskilnik on 05 Oct 2018, 14:13, edited 1 time in total. Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 8255 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: If the integers p, q, r, and s satisfy p<q<r<s, are they consecutive? [#permalink] ### Show Tags 05 Oct 2018, 01:34 => Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution. Since we have 4 variables (p, q, r and s) and 0 equations, E is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first. Conditions 1) & 2) Since $$r – q = 1$$ or $$r = q + 1$$ from condition 1), $$q$$ and $$r$$ are consecutive integers. Since we have $$(p-q)(r-s) = (q-p)(s-r) =1$$, where $$p < q, r < s$$ and $$p-q$$ and $$r-s$$ are integers, we have $$q-p = 1$$ and $$s-r = 1$$, or $$q = p +1$$, and $$s = r +1$$. Thus $$p, q, r$$ and $$s$$ are consecutive integers and both conditions are sufficient when applied together. Since this question is an integer question (one of the key question areas), CMT (Common Mistake Type) 4(A) of the VA (Variable Approach) method tells us that we should also check answers A and B. Condition 1) $$p = 1, q = 2, r = 3$$ and $$s = 4$$ are consecutive integers satisfying $$r – q = 1$$, and the answer is ‘yes’. $$P = 1, q = 2, r = 3$$ and $$s = 5$$ satisfy $$r – q = 1$$, but are not consecutive integers, and the answer is ‘no’. Since we don’t have a unique solution, condition 1) is not sufficient. Condition 2) $$p = 1, q = 2, r = 3$$ and $$s = 4$$ are consecutive integers satisfying $$(p-q)(r-s)=1,$$ and the answer is ‘yes’. $$p = 1, q = 2, r = 4$$ and $$s = 5$$ satisfy $$(p-q)(r-s)=1$$, but are not consecutive integers, and the answer is ‘no’. Since we don’t have a unique solution, condition 2) is not sufficient. Therefore, C is the answer. Answer: C In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions on which the answer is A, B, C or D. _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$79 for 1 month Online Course"
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Re: If the integers p, q, r, and s satisfy p<q<r<s, are they consecutive?   [#permalink] 05 Oct 2018, 01:34
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